File: Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean

full_name: MeasureTheory.«_aux_Mathlib_MeasureTheory_Function_StronglyMeasurable_Basic___macroRules_MeasureTheory_term_→ₛ__1»
code: local infixr:25 " →ₛ " => SimpleFunc
goal: Lean.Macro

full_name: MeasureTheory._aux_Mathlib_MeasureTheory_Function_StronglyMeasurable_Basic___unexpand_MeasureTheory_SimpleFunc_1
code: local infixr:25 " →ₛ " => SimpleFunc
goal: Lean.PrettyPrinter.Unexpander

full_name: MeasureTheory.StronglyMeasurable
code: /-- A function is `StronglyMeasurable` if it is the limit of simple functions. -/
def StronglyMeasurable [MeasurableSpace α] (f : α → β) : Prop 
goal: Prop

full_name: MeasureTheory.«_aux_Mathlib_MeasureTheory_Function_StronglyMeasurable_Basic___macroRules_MeasureTheory_termStronglyMeasurable[_]_1»
code: /-- The notation for StronglyMeasurable giving the measurable space instance explicitly. -/
scoped notation "StronglyMeasurable[" m "]" => @MeasureTheory.StronglyMeasurable _ _ _ m
goal: Lean.Macro

full_name: MeasureTheory.FinStronglyMeasurable
code: /-- A function is `FinStronglyMeasurable` with respect to a measure if it is the limit of simple
  functions with support with finite measure. -/
def FinStronglyMeasurable [Zero β]
    {_ : MeasurableSpace α} (f : α → β) (μ : Measure α := by volume_tac) : Prop 
goal: {x : MeasurableSpace α} → (α → β) → autoParam (MeasureTheory.Measure α) _auto✝ → Prop

full_name: MeasureTheory.AEStronglyMeasurable
code: /-- A function is `AEStronglyMeasurable` with respect to a measure `μ` if it is almost everywhere
equal to the limit of a sequence of simple functions. -/
@[fun_prop]
def AEStronglyMeasurable
    {_ : MeasurableSpace α} (f : α → β) (μ : Measure α := by volume_tac) : Prop 
goal: {x : MeasurableSpace α} → (α → β) → autoParam (MeasureTheory.Measure α) _auto✝ → Prop

full_name: MeasureTheory.AEFinStronglyMeasurable
code: /-- A function is `AEFinStronglyMeasurable` with respect to a measure if it is almost everywhere
equal to the limit of a sequence of simple functions with support with finite measure. -/
def AEFinStronglyMeasurable
    [Zero β] {_ : MeasurableSpace α} (f : α → β) (μ : Measure α := by volume_tac) : Prop 
goal: {x : MeasurableSpace α} → (α → β) → autoParam (MeasureTheory.Measure α) _auto✝ → Prop

full_name: MeasureTheory.StronglyMeasurable.aestronglyMeasurable
code: @[aesop 30% apply (rule_sets := [Measurable])]
protected theorem StronglyMeasurable.aestronglyMeasurable {α β} {_ : MeasurableSpace α}
    [TopologicalSpace β] {f : α → β} {μ : Measure α} (hf : StronglyMeasurable f) :
    AEStronglyMeasurable f μ 
goal: ∀ {x : MeasurableSpace α} [inst : TopologicalSpace β] {f : α → β} {μ : MeasureTheory.Measure α},
  MeasureTheory.StronglyMeasurable f → MeasureTheory.AEStronglyMeasurable f μ

full_name: MeasureTheory.Subsingleton.stronglyMeasurable
code: @[simp]
theorem Subsingleton.stronglyMeasurable {α β} [MeasurableSpace α] [TopologicalSpace β]
    [Subsingleton β] (f : α → β) : StronglyMeasurable f 
goal: MeasureTheory.StronglyMeasurable f

full_name: MeasureTheory.SimpleFunc.stronglyMeasurable
code: theorem SimpleFunc.stronglyMeasurable {α β} {_ : MeasurableSpace α} [TopologicalSpace β]
    (f : α →ₛ β) : StronglyMeasurable f 
goal: ∀ {x : MeasurableSpace α} [inst : TopologicalSpace β] (f : MeasureTheory.SimpleFunc α β),
  MeasureTheory.StronglyMeasurable ↑f

full_name: MeasureTheory.StronglyMeasurable.of_finite
code: @[nontriviality]
theorem StronglyMeasurable.of_finite [Finite α] {_ : MeasurableSpace α}
    [MeasurableSingletonClass α] [TopologicalSpace β]
    (f : α → β) : StronglyMeasurable f 
goal: ∀ {x : MeasurableSpace α} [inst : MeasurableSingletonClass α] [inst : TopologicalSpace β] (f : α → β),
  MeasureTheory.StronglyMeasurable f

full_name: MeasureTheory.stronglyMeasurable_of_isEmpty
code: @[deprecated StronglyMeasurable.of_finite (since := "2024-02-06")]
theorem stronglyMeasurable_of_isEmpty [IsEmpty α] {_ : MeasurableSpace α} [TopologicalSpace β]
    (f : α → β) : StronglyMeasurable f 
goal: ∀ {x : MeasurableSpace α} [inst : TopologicalSpace β] (f : α → β), MeasureTheory.StronglyMeasurable f

full_name: MeasureTheory.stronglyMeasurable_const
code: theorem stronglyMeasurable_const {α β} {_ : MeasurableSpace α} [TopologicalSpace β] {b : β} :
    StronglyMeasurable fun _ : α => b 
goal: ∀ {x : MeasurableSpace α} [inst : TopologicalSpace β] {b : β}, MeasureTheory.StronglyMeasurable fun x => b

full_name: MeasureTheory.stronglyMeasurable_zero
code: @[to_additive]
theorem stronglyMeasurable_one {α β} {_ : MeasurableSpace α} [TopologicalSpace β] [One β] :
    StronglyMeasurable (1 : α → β) 
goal: ∀ {x : MeasurableSpace α} [inst : TopologicalSpace β] [inst_1 : Zero β], MeasureTheory.StronglyMeasurable 0

full_name: MeasureTheory.stronglyMeasurable_one
code: @[to_additive]
theorem stronglyMeasurable_one {α β} {_ : MeasurableSpace α} [TopologicalSpace β] [One β] :
    StronglyMeasurable (1 : α → β) 
goal: ∀ {x : MeasurableSpace α} [inst : TopologicalSpace β] [inst_1 : One β], MeasureTheory.StronglyMeasurable 1

full_name: MeasureTheory.stronglyMeasurable_const'
code: /-- A version of `stronglyMeasurable_const` that assumes `f x = f y` for all `x, y`.
This version works for functions between empty types. -/
theorem stronglyMeasurable_const' {α β} {m : MeasurableSpace α} [TopologicalSpace β] {f : α → β}
    (hf : ∀ x y, f x = f y) : StronglyMeasurable f 
goal: MeasureTheory.StronglyMeasurable f

full_name: MeasureTheory.Subsingleton.stronglyMeasurable'
code: @[simp]
theorem Subsingleton.stronglyMeasurable' {α β} [MeasurableSpace α] [TopologicalSpace β]
    [Subsingleton α] (f : α → β) : StronglyMeasurable f 
goal: MeasureTheory.StronglyMeasurable f

full_name: MeasureTheory.StronglyMeasurable.approx
code: /-- A sequence of simple functions such that
`∀ x, Tendsto (fun n => hf.approx n x) atTop (𝓝 (f x))`.
That property is given by `stronglyMeasurable.tendsto_approx`. -/
protected noncomputable def approx {_ : MeasurableSpace α} (hf : StronglyMeasurable f) :
    ℕ → α →ₛ β 
goal: {x : MeasurableSpace α} → MeasureTheory.StronglyMeasurable f → ℕ → MeasureTheory.SimpleFunc α β

full_name: MeasureTheory.StronglyMeasurable.tendsto_approx
code: protected theorem tendsto_approx {_ : MeasurableSpace α} (hf : StronglyMeasurable f) :
    ∀ x, Tendsto (fun n => hf.approx n x) atTop (𝓝 (f x)) 
goal: ∀ {x : MeasurableSpace α} (hf : MeasureTheory.StronglyMeasurable f) (x_1 : α),
  Filter.Tendsto (fun n => ↑(hf.approx n) x_1) Filter.atTop (𝓝 (f x_1))

full_name: MeasureTheory.StronglyMeasurable.approxBounded
code: /-- Similar to `stronglyMeasurable.approx`, but enforces that the norm of every function in the
sequence is less than `c` everywhere. If `‖f x‖ ≤ c` this sequence of simple functions verifies
`Tendsto (fun n => hf.approxBounded n x) atTop (𝓝 (f x))`. -/
noncomputable def approxBounded {_ : MeasurableSpace α} [Norm β] [SMul ℝ β]
    (hf : StronglyMeasurable f) (c : ℝ) : ℕ → SimpleFunc α β 
goal: {x : MeasurableSpace α} →
  [inst : Norm β] → [inst : SMul ℝ β] → MeasureTheory.StronglyMeasurable f → ℝ → ℕ → MeasureTheory.SimpleFunc α β

full_name: MeasureTheory.StronglyMeasurable.tendsto_approxBounded_of_norm_le
code: theorem tendsto_approxBounded_of_norm_le {β} {f : α → β} [NormedAddCommGroup β] [NormedSpace ℝ β]
    {m : MeasurableSpace α} (hf : StronglyMeasurable[m] f) {c : ℝ} {x : α} (hfx : ‖f x‖ ≤ c) :
    Tendsto (fun n => hf.approxBounded c n x) atTop (𝓝 (f x)) 
goal: Filter.Tendsto (fun n => ↑(hf.approxBounded c n) x) Filter.atTop (𝓝 (f x))

full_name: MeasureTheory.StronglyMeasurable.tendsto_approxBounded_ae
code: theorem tendsto_approxBounded_ae {β} {f : α → β} [NormedAddCommGroup β] [NormedSpace ℝ β]
    {m m0 : MeasurableSpace α} {μ : Measure α} (hf : StronglyMeasurable[m] f) {c : ℝ}
    (hf_bound : ∀ᵐ x ∂μ, ‖f x‖ ≤ c) :
    ∀ᵐ x ∂μ, Tendsto (fun n => hf.approxBounded c n x) atTop (𝓝 (f x)) 
goal: ∀ᵐ (x : α) ∂μ, Filter.Tendsto (fun n => ↑(hf.approxBounded c n) x) Filter.atTop (𝓝 (f x))

full_name: MeasureTheory.StronglyMeasurable.norm_approxBounded_le
code: theorem norm_approxBounded_le {β} {f : α → β} [SeminormedAddCommGroup β] [NormedSpace ℝ β]
    {m : MeasurableSpace α} {c : ℝ} (hf : StronglyMeasurable[m] f) (hc : 0 ≤ c) (n : ℕ) (x : α) :
    ‖hf.approxBounded c n x‖ ≤ c 
goal: ‖↑(hf.approxBounded c n) x‖ ≤ c

full_name: stronglyMeasurable_bot_iff
code: theorem _root_.stronglyMeasurable_bot_iff [Nonempty β] [T2Space β] :
    StronglyMeasurable[⊥] f ↔ ∃ c, f = fun _ => c 
goal: MeasureTheory.StronglyMeasurable f ↔ ∃ c, f = fun x => c

full_name: MeasureTheory.StronglyMeasurable.finStronglyMeasurable_of_set_sigmaFinite
code: theorem finStronglyMeasurable_of_set_sigmaFinite [TopologicalSpace β] [Zero β]
    {m : MeasurableSpace α} {μ : Measure α} (hf_meas : StronglyMeasurable f) {t : Set α}
    (ht : MeasurableSet t) (hft_zero : ∀ x ∈ tᶜ, f x = 0) (htμ : SigmaFinite (μ.restrict t)) :
    FinStronglyMeasurable f μ 
goal: MeasureTheory.FinStronglyMeasurable f μ

full_name: MeasureTheory.StronglyMeasurable.finStronglyMeasurable
code: /-- If the measure is sigma-finite, all strongly measurable functions are
  `FinStronglyMeasurable`. -/
@[aesop 5% apply (rule_sets := [Measurable])]
protected theorem finStronglyMeasurable [TopologicalSpace β] [Zero β] {m0 : MeasurableSpace α}
    (hf : StronglyMeasurable f) (μ : Measure α) [SigmaFinite μ] : FinStronglyMeasurable f μ 
goal: MeasureTheory.FinStronglyMeasurable f μ

full_name: MeasureTheory.StronglyMeasurable.measurable
code: /-- A strongly measurable function is measurable. -/
@[aesop 5% apply (rule_sets := [Measurable])]
protected theorem measurable {_ : MeasurableSpace α} [TopologicalSpace β] [PseudoMetrizableSpace β]
    [MeasurableSpace β] [BorelSpace β] (hf : StronglyMeasurable f) : Measurable f 
goal: ∀ {x : MeasurableSpace α} [inst : TopologicalSpace β] [inst_1 : TopologicalSpace.PseudoMetrizableSpace β]
  [inst_2 : MeasurableSpace β] [inst_3 : BorelSpace β], MeasureTheory.StronglyMeasurable f → Measurable f

full_name: MeasureTheory.StronglyMeasurable.aemeasurable
code: /-- A strongly measurable function is almost everywhere measurable. -/
@[aesop 5% apply (rule_sets := [Measurable])]
protected theorem aemeasurable {_ : MeasurableSpace α} [TopologicalSpace β]
    [PseudoMetrizableSpace β] [MeasurableSpace β] [BorelSpace β] {μ : Measure α}
    (hf : StronglyMeasurable f) : AEMeasurable f μ 
goal: ∀ {x : MeasurableSpace α} [inst : TopologicalSpace β] [inst_1 : TopologicalSpace.PseudoMetrizableSpace β]
  [inst_2 : MeasurableSpace β] [inst_3 : BorelSpace β] {μ : MeasureTheory.Measure α},
  MeasureTheory.StronglyMeasurable f → AEMeasurable f μ

full_name: Continuous.comp_stronglyMeasurable
code: theorem _root_.Continuous.comp_stronglyMeasurable {_ : MeasurableSpace α} [TopologicalSpace β]
    [TopologicalSpace γ] {g : β → γ} {f : α → β} (hg : Continuous g) (hf : StronglyMeasurable f) :
    StronglyMeasurable fun x => g (f x) 
goal: ∀ {x : MeasurableSpace α} [inst : TopologicalSpace β] [inst_1 : TopologicalSpace γ] {g : β → γ} {f : α → β},
  Continuous g → MeasureTheory.StronglyMeasurable f → MeasureTheory.StronglyMeasurable fun x => g (f x)

full_name: MeasureTheory.StronglyMeasurable.measurableSet_support
code: @[to_additive]
nonrec theorem measurableSet_mulSupport {m : MeasurableSpace α} [One β] [TopologicalSpace β]
    [MetrizableSpace β] (hf : StronglyMeasurable f) : MeasurableSet (mulSupport f) 
goal: MeasurableSet (Function.support f)

full_name: MeasureTheory.StronglyMeasurable.measurableSet_mulSupport
code: @[to_additive]
nonrec theorem measurableSet_mulSupport {m : MeasurableSpace α} [One β] [TopologicalSpace β]
    [MetrizableSpace β] (hf : StronglyMeasurable f) : MeasurableSet (mulSupport f) 
goal: MeasurableSet (Function.mulSupport f)

full_name: MeasureTheory.StronglyMeasurable.mono
code: protected theorem mono {m m' : MeasurableSpace α} [TopologicalSpace β]
    (hf : StronglyMeasurable[m'] f) (h_mono : m' ≤ m) : StronglyMeasurable[m] f 
goal: MeasureTheory.StronglyMeasurable f

full_name: MeasureTheory.StronglyMeasurable.prod_mk
code: protected theorem prod_mk {m : MeasurableSpace α} [TopologicalSpace β] [TopologicalSpace γ]
    {f : α → β} {g : α → γ} (hf : StronglyMeasurable f) (hg : StronglyMeasurable g) :
    StronglyMeasurable fun x => (f x, g x) 
goal: MeasureTheory.StronglyMeasurable fun x => (f x, g x)

full_name: MeasureTheory.StronglyMeasurable.comp_measurable
code: theorem comp_measurable [TopologicalSpace β] {_ : MeasurableSpace α} {_ : MeasurableSpace γ}
    {f : α → β} {g : γ → α} (hf : StronglyMeasurable f) (hg : Measurable g) :
    StronglyMeasurable (f ∘ g) 
goal: ∀ {x : MeasurableSpace α} {x_1 : MeasurableSpace γ} {f : α → β} {g : γ → α},
  MeasureTheory.StronglyMeasurable f → Measurable g → MeasureTheory.StronglyMeasurable (f ∘ g)

full_name: MeasureTheory.StronglyMeasurable.of_uncurry_left
code: theorem of_uncurry_left [TopologicalSpace β] {_ : MeasurableSpace α} {_ : MeasurableSpace γ}
    {f : α → γ → β} (hf : StronglyMeasurable (uncurry f)) {x : α} : StronglyMeasurable (f x) 
goal: ∀ {x : MeasurableSpace α} {x_1 : MeasurableSpace γ} {f : α → γ → β},
  MeasureTheory.StronglyMeasurable (Function.uncurry f) → ∀ {x : α}, MeasureTheory.StronglyMeasurable (f x)

full_name: MeasureTheory.StronglyMeasurable.of_uncurry_right
code: theorem of_uncurry_right [TopologicalSpace β] {_ : MeasurableSpace α} {_ : MeasurableSpace γ}
    {f : α → γ → β} (hf : StronglyMeasurable (uncurry f)) {y : γ} :
    StronglyMeasurable fun x => f x y 
goal: ∀ {x : MeasurableSpace α} {x_1 : MeasurableSpace γ} {f : α → γ → β},
  MeasureTheory.StronglyMeasurable (Function.uncurry f) → ∀ {y : γ}, MeasureTheory.StronglyMeasurable fun x => f x y

full_name: MeasureTheory.StronglyMeasurable.add
code: @[to_additive (attr := aesop safe 20 apply (rule_sets := [Measurable]))]
protected theorem mul [Mul β] [ContinuousMul β] (hf : StronglyMeasurable f)
    (hg : StronglyMeasurable g) : StronglyMeasurable (f * g) 
goal: MeasureTheory.StronglyMeasurable (f + g)

full_name: MeasureTheory.StronglyMeasurable.mul
code: @[to_additive (attr := aesop safe 20 apply (rule_sets := [Measurable]))]
protected theorem mul [Mul β] [ContinuousMul β] (hf : StronglyMeasurable f)
    (hg : StronglyMeasurable g) : StronglyMeasurable (f * g) 
goal: MeasureTheory.StronglyMeasurable (f * g)

full_name: MeasureTheory.StronglyMeasurable.add_const
code: @[to_additive (attr := measurability)]
theorem mul_const [Mul β] [ContinuousMul β] (hf : StronglyMeasurable f) (c : β) :
    StronglyMeasurable fun x => f x * c 
goal: MeasureTheory.StronglyMeasurable fun x => f x + c

full_name: MeasureTheory.StronglyMeasurable.mul_const
code: @[to_additive (attr := measurability)]
theorem mul_const [Mul β] [ContinuousMul β] (hf : StronglyMeasurable f) (c : β) :
    StronglyMeasurable fun x => f x * c 
goal: MeasureTheory.StronglyMeasurable fun x => f x * c

full_name: MeasureTheory.StronglyMeasurable.const_mul
code: @[to_additive (attr := measurability)]
theorem const_mul [Mul β] [ContinuousMul β] (hf : StronglyMeasurable f) (c : β) :
    StronglyMeasurable fun x => c * f x 
goal: MeasureTheory.StronglyMeasurable fun x => c * f x

full_name: MeasureTheory.StronglyMeasurable.const_add
code: @[to_additive (attr := measurability)]
theorem const_mul [Mul β] [ContinuousMul β] (hf : StronglyMeasurable f) (c : β) :
    StronglyMeasurable fun x => c * f x 
goal: MeasureTheory.StronglyMeasurable fun x => c + f x

full_name: MeasureTheory.StronglyMeasurable.pow
code: @[to_additive (attr := aesop safe 20 apply (rule_sets := [Measurable])) const_nsmul]
protected theorem pow [Monoid β] [ContinuousMul β] (hf : StronglyMeasurable f) (n : ℕ) :
    StronglyMeasurable (f ^ n) 
goal: MeasureTheory.StronglyMeasurable (f ^ n)

full_name: MeasureTheory.StronglyMeasurable.const_nsmul
code: @[to_additive (attr := aesop safe 20 apply (rule_sets := [Measurable])) const_nsmul]
protected theorem pow [Monoid β] [ContinuousMul β] (hf : StronglyMeasurable f) (n : ℕ) :
    StronglyMeasurable (f ^ n) 
goal: MeasureTheory.StronglyMeasurable (n • f)

full_name: MeasureTheory.StronglyMeasurable.inv
code: @[to_additive (attr := measurability)]
protected theorem inv [Inv β] [ContinuousInv β] (hf : StronglyMeasurable f) :
    StronglyMeasurable f⁻¹ 
goal: MeasureTheory.StronglyMeasurable f⁻¹

full_name: MeasureTheory.StronglyMeasurable.neg
code: @[to_additive (attr := measurability)]
protected theorem inv [Inv β] [ContinuousInv β] (hf : StronglyMeasurable f) :
    StronglyMeasurable f⁻¹ 
goal: MeasureTheory.StronglyMeasurable (-f)

full_name: MeasureTheory.StronglyMeasurable.div
code: @[to_additive (attr := aesop safe 20 apply (rule_sets := [Measurable]))]
protected theorem div [Div β] [ContinuousDiv β] (hf : StronglyMeasurable f)
    (hg : StronglyMeasurable g) : StronglyMeasurable (f / g) 
goal: MeasureTheory.StronglyMeasurable (f / g)

full_name: MeasureTheory.StronglyMeasurable.sub
code: @[to_additive (attr := aesop safe 20 apply (rule_sets := [Measurable]))]
protected theorem div [Div β] [ContinuousDiv β] (hf : StronglyMeasurable f)
    (hg : StronglyMeasurable g) : StronglyMeasurable (f / g) 
goal: MeasureTheory.StronglyMeasurable (f - g)

full_name: MeasureTheory.StronglyMeasurable.mul_iff_right
code: @[to_additive]
theorem mul_iff_right [CommGroup β] [TopologicalGroup β] (hf : StronglyMeasurable f) :
    StronglyMeasurable (f * g) ↔ StronglyMeasurable g 
goal: MeasureTheory.StronglyMeasurable (f * g) ↔ MeasureTheory.StronglyMeasurable g

full_name: MeasureTheory.StronglyMeasurable.add_iff_right
code: @[to_additive]
theorem mul_iff_right [CommGroup β] [TopologicalGroup β] (hf : StronglyMeasurable f) :
    StronglyMeasurable (f * g) ↔ StronglyMeasurable g 
goal: MeasureTheory.StronglyMeasurable (f + g) ↔ MeasureTheory.StronglyMeasurable g

full_name: MeasureTheory.StronglyMeasurable.mul_iff_left
code: @[to_additive]
theorem mul_iff_left [CommGroup β] [TopologicalGroup β] (hf : StronglyMeasurable f) :
    StronglyMeasurable (g * f) ↔ StronglyMeasurable g 
goal: MeasureTheory.StronglyMeasurable (g * f) ↔ MeasureTheory.StronglyMeasurable g

full_name: MeasureTheory.StronglyMeasurable.add_iff_left
code: @[to_additive]
theorem mul_iff_left [CommGroup β] [TopologicalGroup β] (hf : StronglyMeasurable f) :
    StronglyMeasurable (g * f) ↔ StronglyMeasurable g 
goal: MeasureTheory.StronglyMeasurable (g + f) ↔ MeasureTheory.StronglyMeasurable g

full_name: MeasureTheory.StronglyMeasurable.vadd
code: @[to_additive (attr := aesop safe 20 apply (rule_sets := [Measurable]))]
protected theorem smul {𝕜} [TopologicalSpace 𝕜] [SMul 𝕜 β] [ContinuousSMul 𝕜 β] {f : α → 𝕜}
    {g : α → β} (hf : StronglyMeasurable f) (hg : StronglyMeasurable g) :
    StronglyMeasurable fun x => f x • g x 
goal: MeasureTheory.StronglyMeasurable fun x => f x +ᵥ g x

full_name: MeasureTheory.StronglyMeasurable.smul
code: @[to_additive (attr := aesop safe 20 apply (rule_sets := [Measurable]))]
protected theorem smul {𝕜} [TopologicalSpace 𝕜] [SMul 𝕜 β] [ContinuousSMul 𝕜 β] {f : α → 𝕜}
    {g : α → β} (hf : StronglyMeasurable f) (hg : StronglyMeasurable g) :
    StronglyMeasurable fun x => f x • g x 
goal: MeasureTheory.StronglyMeasurable fun x => f x • g x

full_name: MeasureTheory.StronglyMeasurable.const_smul
code: @[to_additive (attr := measurability)]
protected theorem const_smul {𝕜} [SMul 𝕜 β] [ContinuousConstSMul 𝕜 β] (hf : StronglyMeasurable f)
    (c : 𝕜) : StronglyMeasurable (c • f) 
goal: MeasureTheory.StronglyMeasurable (c • f)

full_name: MeasureTheory.StronglyMeasurable.const_vadd
code: @[to_additive (attr := measurability)]
protected theorem const_smul {𝕜} [SMul 𝕜 β] [ContinuousConstSMul 𝕜 β] (hf : StronglyMeasurable f)
    (c : 𝕜) : StronglyMeasurable (c • f) 
goal: MeasureTheory.StronglyMeasurable (c +ᵥ f)

full_name: MeasureTheory.StronglyMeasurable.const_smul'
code: @[to_additive (attr := measurability)]
protected theorem const_smul' {𝕜} [SMul 𝕜 β] [ContinuousConstSMul 𝕜 β] (hf : StronglyMeasurable f)
    (c : 𝕜) : StronglyMeasurable fun x => c • f x 
goal: MeasureTheory.StronglyMeasurable fun x => c • f x

full_name: MeasureTheory.StronglyMeasurable.const_vadd'
code: @[to_additive (attr := measurability)]
protected theorem const_smul' {𝕜} [SMul 𝕜 β] [ContinuousConstSMul 𝕜 β] (hf : StronglyMeasurable f)
    (c : 𝕜) : StronglyMeasurable fun x => c • f x 
goal: MeasureTheory.StronglyMeasurable fun x => c +ᵥ f x

full_name: MeasureTheory.StronglyMeasurable.vadd_const
code: @[to_additive (attr := measurability)]
protected theorem smul_const {𝕜} [TopologicalSpace 𝕜] [SMul 𝕜 β] [ContinuousSMul 𝕜 β] {f : α → 𝕜}
    (hf : StronglyMeasurable f) (c : β) : StronglyMeasurable fun x => f x • c 
goal: MeasureTheory.StronglyMeasurable fun x => f x +ᵥ c

full_name: MeasureTheory.StronglyMeasurable.smul_const
code: @[to_additive (attr := measurability)]
protected theorem smul_const {𝕜} [TopologicalSpace 𝕜] [SMul 𝕜 β] [ContinuousSMul 𝕜 β] {f : α → 𝕜}
    (hf : StronglyMeasurable f) (c : β) : StronglyMeasurable fun x => f x • c 
goal: MeasureTheory.StronglyMeasurable fun x => f x • c

full_name: Measurable.add_stronglyMeasurable
code: /-- In a normed vector space, the addition of a measurable function and a strongly measurable
function is measurable. Note that this is not true without further second-countability assumptions
for the addition of two measurable functions. -/
theorem _root_.Measurable.add_stronglyMeasurable
    {α E : Type*} {_ : MeasurableSpace α} [AddGroup E] [TopologicalSpace E]
    [MeasurableSpace E] [BorelSpace E] [ContinuousAdd E] [PseudoMetrizableSpace E]
    {g f : α → E} (hg : Measurable g) (hf : StronglyMeasurable f) :
    Measurable (g + f) 
goal: ∀ {x : MeasurableSpace α} [inst : AddGroup E] [inst_1 : TopologicalSpace E] [inst_2 : MeasurableSpace E]
  [inst_3 : BorelSpace E] [inst_4 : ContinuousAdd E] [inst_5 : TopologicalSpace.PseudoMetrizableSpace E] {g f : α → E},
  Measurable g → MeasureTheory.StronglyMeasurable f → Measurable (g + f)

full_name: Measurable.sub_stronglyMeasurable
code: /-- In a normed vector space, the subtraction of a measurable function and a strongly measurable
function is measurable. Note that this is not true without further second-countability assumptions
for the subtraction of two measurable functions. -/
theorem _root_.Measurable.sub_stronglyMeasurable
    {α E : Type*} {_ : MeasurableSpace α} [AddCommGroup E] [TopologicalSpace E]
    [MeasurableSpace E] [BorelSpace E] [ContinuousAdd E] [ContinuousNeg E] [PseudoMetrizableSpace E]
    {g f : α → E} (hg : Measurable g) (hf : StronglyMeasurable f) :
    Measurable (g - f) 
goal: ∀ {x : MeasurableSpace α} [inst : AddCommGroup E] [inst_1 : TopologicalSpace E] [inst_2 : MeasurableSpace E]
  [inst_3 : BorelSpace E] [inst_4 : ContinuousAdd E] [inst_5 : ContinuousNeg E]
  [inst_6 : TopologicalSpace.PseudoMetrizableSpace E] {g f : α → E},
  Measurable g → MeasureTheory.StronglyMeasurable f → Measurable (g - f)

full_name: Measurable.stronglyMeasurable_add
code: /-- In a normed vector space, the addition of a strongly measurable function and a measurable
function is measurable. Note that this is not true without further second-countability assumptions
for the addition of two measurable functions. -/
theorem _root_.Measurable.stronglyMeasurable_add
    {α E : Type*} {_ : MeasurableSpace α} [AddGroup E] [TopologicalSpace E]
    [MeasurableSpace E] [BorelSpace E] [ContinuousAdd E] [PseudoMetrizableSpace E]
    {g f : α → E} (hg : Measurable g) (hf : StronglyMeasurable f) :
    Measurable (f + g) 
goal: ∀ {x : MeasurableSpace α} [inst : AddGroup E] [inst_1 : TopologicalSpace E] [inst_2 : MeasurableSpace E]
  [inst_3 : BorelSpace E] [inst_4 : ContinuousAdd E] [inst_5 : TopologicalSpace.PseudoMetrizableSpace E] {g f : α → E},
  Measurable g → MeasureTheory.StronglyMeasurable f → Measurable (f + g)

full_name: stronglyMeasurable_const_smul_iff
code: theorem _root_.stronglyMeasurable_const_smul_iff {m : MeasurableSpace α} (c : G) :
    (StronglyMeasurable fun x => c • f x) ↔ StronglyMeasurable f 
goal: (MeasureTheory.StronglyMeasurable fun x => c • f x) ↔ MeasureTheory.StronglyMeasurable f

full_name: IsUnit.stronglyMeasurable_const_smul_iff
code: nonrec theorem _root_.IsUnit.stronglyMeasurable_const_smul_iff {_ : MeasurableSpace α} {c : M}
    (hc : IsUnit c) :
    (StronglyMeasurable fun x => c • f x) ↔ StronglyMeasurable f 
goal: ∀ {x : MeasurableSpace α} {c : M},
  IsUnit c → ((MeasureTheory.StronglyMeasurable fun x => c • f x) ↔ MeasureTheory.StronglyMeasurable f)

full_name: stronglyMeasurable_const_smul_iff₀
code: theorem _root_.stronglyMeasurable_const_smul_iff₀ {_ : MeasurableSpace α} {c : G₀} (hc : c ≠ 0) :
    (StronglyMeasurable fun x => c • f x) ↔ StronglyMeasurable f 
goal: ∀ {x : MeasurableSpace α} {c : G₀},
  c ≠ 0 → ((MeasureTheory.StronglyMeasurable fun x => c • f x) ↔ MeasureTheory.StronglyMeasurable f)

full_name: MeasureTheory.StronglyMeasurable.sup
code: @[aesop safe 20 (rule_sets := [Measurable])]
protected theorem sup [Sup β] [ContinuousSup β] (hf : StronglyMeasurable f)
    (hg : StronglyMeasurable g) : StronglyMeasurable (f ⊔ g) 
goal: MeasureTheory.StronglyMeasurable (f ⊔ g)

full_name: MeasureTheory.StronglyMeasurable.inf
code: @[aesop safe 20 (rule_sets := [Measurable])]
protected theorem inf [Inf β] [ContinuousInf β] (hf : StronglyMeasurable f)
    (hg : StronglyMeasurable g) : StronglyMeasurable (f ⊓ g) 
goal: MeasureTheory.StronglyMeasurable (f ⊓ g)

full_name: List.stronglyMeasurable_sum'
code: @[to_additive (attr := measurability)]
theorem _root_.List.stronglyMeasurable_prod' (l : List (α → M))
    (hl : ∀ f ∈ l, StronglyMeasurable f) : StronglyMeasurable l.prod 
goal: MeasureTheory.StronglyMeasurable l.sum

full_name: List.stronglyMeasurable_prod'
code: @[to_additive (attr := measurability)]
theorem _root_.List.stronglyMeasurable_prod' (l : List (α → M))
    (hl : ∀ f ∈ l, StronglyMeasurable f) : StronglyMeasurable l.prod 
goal: MeasureTheory.StronglyMeasurable l.prod

full_name: List.stronglyMeasurable_prod
code: @[to_additive (attr := measurability)]
theorem _root_.List.stronglyMeasurable_prod (l : List (α → M))
    (hl : ∀ f ∈ l, StronglyMeasurable f) :
    StronglyMeasurable fun x => (l.map fun f : α → M => f x).prod 
goal: MeasureTheory.StronglyMeasurable fun x => (List.map (fun f => f x) l).prod

full_name: List.stronglyMeasurable_sum
code: @[to_additive (attr := measurability)]
theorem _root_.List.stronglyMeasurable_prod (l : List (α → M))
    (hl : ∀ f ∈ l, StronglyMeasurable f) :
    StronglyMeasurable fun x => (l.map fun f : α → M => f x).prod 
goal: MeasureTheory.StronglyMeasurable fun x => (List.map (fun f => f x) l).sum

full_name: Multiset.stronglyMeasurable_prod'
code: @[to_additive (attr := measurability)]
theorem _root_.Multiset.stronglyMeasurable_prod' (l : Multiset (α → M))
    (hl : ∀ f ∈ l, StronglyMeasurable f) : StronglyMeasurable l.prod 
goal: MeasureTheory.StronglyMeasurable l.prod

full_name: Multiset.stronglyMeasurable_sum'
code: @[to_additive (attr := measurability)]
theorem _root_.Multiset.stronglyMeasurable_prod' (l : Multiset (α → M))
    (hl : ∀ f ∈ l, StronglyMeasurable f) : StronglyMeasurable l.prod 
goal: MeasureTheory.StronglyMeasurable l.sum

full_name: Multiset.stronglyMeasurable_sum
code: @[to_additive (attr := measurability)]
theorem _root_.Multiset.stronglyMeasurable_prod (s : Multiset (α → M))
    (hs : ∀ f ∈ s, StronglyMeasurable f) :
    StronglyMeasurable fun x => (s.map fun f : α → M => f x).prod 
goal: MeasureTheory.StronglyMeasurable fun x => (Multiset.map (fun f => f x) s).sum

full_name: Multiset.stronglyMeasurable_prod
code: @[to_additive (attr := measurability)]
theorem _root_.Multiset.stronglyMeasurable_prod (s : Multiset (α → M))
    (hs : ∀ f ∈ s, StronglyMeasurable f) :
    StronglyMeasurable fun x => (s.map fun f : α → M => f x).prod 
goal: MeasureTheory.StronglyMeasurable fun x => (Multiset.map (fun f => f x) s).prod

full_name: Finset.stronglyMeasurable_sum'
code: @[to_additive (attr := measurability)]
theorem _root_.Finset.stronglyMeasurable_prod' {ι : Type*} {f : ι → α → M} (s : Finset ι)
    (hf : ∀ i ∈ s, StronglyMeasurable (f i)) : StronglyMeasurable (∏ i ∈ s, f i) 
goal: MeasureTheory.StronglyMeasurable (∑ i ∈ s, f i)

full_name: Finset.stronglyMeasurable_prod'
code: @[to_additive (attr := measurability)]
theorem _root_.Finset.stronglyMeasurable_prod' {ι : Type*} {f : ι → α → M} (s : Finset ι)
    (hf : ∀ i ∈ s, StronglyMeasurable (f i)) : StronglyMeasurable (∏ i ∈ s, f i) 
goal: MeasureTheory.StronglyMeasurable (∏ i ∈ s, f i)

full_name: Finset.stronglyMeasurable_sum
code: @[to_additive (attr := measurability)]
theorem _root_.Finset.stronglyMeasurable_prod {ι : Type*} {f : ι → α → M} (s : Finset ι)
    (hf : ∀ i ∈ s, StronglyMeasurable (f i)) : StronglyMeasurable fun a => ∏ i ∈ s, f i a 
goal: MeasureTheory.StronglyMeasurable fun a => ∑ i ∈ s, f i a

full_name: Finset.stronglyMeasurable_prod
code: @[to_additive (attr := measurability)]
theorem _root_.Finset.stronglyMeasurable_prod {ι : Type*} {f : ι → α → M} (s : Finset ι)
    (hf : ∀ i ∈ s, StronglyMeasurable (f i)) : StronglyMeasurable fun a => ∏ i ∈ s, f i a 
goal: MeasureTheory.StronglyMeasurable fun a => ∏ i ∈ s, f i a

full_name: MeasureTheory.StronglyMeasurable.isSeparable_range
code: /-- The range of a strongly measurable function is separable. -/
protected theorem isSeparable_range {m : MeasurableSpace α} [TopologicalSpace β]
    (hf : StronglyMeasurable f) : TopologicalSpace.IsSeparable (range f) 
goal: TopologicalSpace.IsSeparable (Set.range f)

full_name: MeasureTheory.StronglyMeasurable.separableSpace_range_union_singleton
code: theorem separableSpace_range_union_singleton {_ : MeasurableSpace α} [TopologicalSpace β]
    [PseudoMetrizableSpace β] (hf : StronglyMeasurable f) {b : β} :
    SeparableSpace (range f ∪ {b} : Set β) 
goal: ∀ {x : MeasurableSpace α} [inst : TopologicalSpace β] [inst_1 : TopologicalSpace.PseudoMetrizableSpace β],
  MeasureTheory.StronglyMeasurable f → ∀ {b : β}, TopologicalSpace.SeparableSpace ↑(Set.range f ∪ {b})

full_name: Measurable.stronglyMeasurable
code: /-- In a space with second countable topology, measurable implies strongly measurable. -/
@[aesop 90% apply (rule_sets := [Measurable])]
theorem _root_.Measurable.stronglyMeasurable [TopologicalSpace β] [PseudoMetrizableSpace β]
    [SecondCountableTopology β] [OpensMeasurableSpace β] (hf : Measurable f) :
    StronglyMeasurable f 
goal: MeasureTheory.StronglyMeasurable f

full_name: stronglyMeasurable_iff_measurable
code: /-- In a space with second countable topology, strongly measurable and measurable are equivalent. -/
theorem _root_.stronglyMeasurable_iff_measurable [TopologicalSpace β] [MetrizableSpace β]
    [BorelSpace β] [SecondCountableTopology β] : StronglyMeasurable f ↔ Measurable f 
goal: MeasureTheory.StronglyMeasurable f ↔ Measurable f

full_name: stronglyMeasurable_id
code: @[measurability]
theorem _root_.stronglyMeasurable_id [TopologicalSpace α] [PseudoMetrizableSpace α]
    [OpensMeasurableSpace α] [SecondCountableTopology α] : StronglyMeasurable (id : α → α) 
goal: MeasureTheory.StronglyMeasurable id

full_name: stronglyMeasurable_iff_measurable_separable
code: /-- A function is strongly measurable if and only if it is measurable and has separable
range. -/
theorem _root_.stronglyMeasurable_iff_measurable_separable {m : MeasurableSpace α}
    [TopologicalSpace β] [PseudoMetrizableSpace β] [MeasurableSpace β] [BorelSpace β] :
    StronglyMeasurable f ↔ Measurable f ∧ IsSeparable (range f) 
goal: MeasureTheory.StronglyMeasurable f ↔ Measurable f ∧ TopologicalSpace.IsSeparable (Set.range f)

full_name: Continuous.stronglyMeasurable
code: /-- A continuous function is strongly measurable when either the source space or the target space
is second-countable. -/
theorem _root_.Continuous.stronglyMeasurable [MeasurableSpace α] [TopologicalSpace α]
    [OpensMeasurableSpace α] [TopologicalSpace β] [PseudoMetrizableSpace β]
    [h : SecondCountableTopologyEither α β] {f : α → β} (hf : Continuous f) :
    StronglyMeasurable f 
goal: MeasureTheory.StronglyMeasurable f

full_name: Continuous.stronglyMeasurable_of_support_subset_isCompact
code: /-- A continuous function whose support is contained in a compact set is strongly measurable. -/
@[to_additive]
theorem _root_.Continuous.stronglyMeasurable_of_mulSupport_subset_isCompact
    [MeasurableSpace α] [TopologicalSpace α] [OpensMeasurableSpace α] [MeasurableSpace β]
    [TopologicalSpace β] [PseudoMetrizableSpace β] [BorelSpace β] [One β] {f : α → β}
    (hf : Continuous f) {k : Set α} (hk : IsCompact k)
    (h'f : mulSupport f ⊆ k) : StronglyMeasurable f 
goal: MeasureTheory.StronglyMeasurable f

full_name: Continuous.stronglyMeasurable_of_mulSupport_subset_isCompact
code: /-- A continuous function whose support is contained in a compact set is strongly measurable. -/
@[to_additive]
theorem _root_.Continuous.stronglyMeasurable_of_mulSupport_subset_isCompact
    [MeasurableSpace α] [TopologicalSpace α] [OpensMeasurableSpace α] [MeasurableSpace β]
    [TopologicalSpace β] [PseudoMetrizableSpace β] [BorelSpace β] [One β] {f : α → β}
    (hf : Continuous f) {k : Set α} (hk : IsCompact k)
    (h'f : mulSupport f ⊆ k) : StronglyMeasurable f 
goal: MeasureTheory.StronglyMeasurable f

full_name: Continuous.stronglyMeasurable_of_hasCompactMulSupport
code: /-- A continuous function with compact support is strongly measurable. -/
@[to_additive]
theorem _root_.Continuous.stronglyMeasurable_of_hasCompactMulSupport
    [MeasurableSpace α] [TopologicalSpace α] [OpensMeasurableSpace α] [MeasurableSpace β]
    [TopologicalSpace β] [PseudoMetrizableSpace β] [BorelSpace β] [One β] {f : α → β}
    (hf : Continuous f) (h'f : HasCompactMulSupport f) : StronglyMeasurable f 
goal: MeasureTheory.StronglyMeasurable f

full_name: Continuous.stronglyMeasurable_of_hasCompactSupport
code: /-- A continuous function with compact support is strongly measurable. -/
@[to_additive]
theorem _root_.Continuous.stronglyMeasurable_of_hasCompactMulSupport
    [MeasurableSpace α] [TopologicalSpace α] [OpensMeasurableSpace α] [MeasurableSpace β]
    [TopologicalSpace β] [PseudoMetrizableSpace β] [BorelSpace β] [One β] {f : α → β}
    (hf : Continuous f) (h'f : HasCompactMulSupport f) : StronglyMeasurable f 
goal: MeasureTheory.StronglyMeasurable f

full_name: HasCompactSupport.stronglyMeasurable_of_prod
code: /-- A continuous function with compact support on a product space is strongly measurable for the
product sigma-algebra. The subtlety is that we do not assume that the spaces are separable, so the
product of the Borel sigma algebras might not contain all open sets, but still it contains enough
of them to approximate compactly supported continuous functions. -/
lemma _root_.HasCompactSupport.stronglyMeasurable_of_prod {X Y : Type*} [Zero α]
    [TopologicalSpace X] [TopologicalSpace Y] [MeasurableSpace X] [MeasurableSpace Y]
    [OpensMeasurableSpace X] [OpensMeasurableSpace Y] [TopologicalSpace α] [PseudoMetrizableSpace α]
    {f : X × Y → α} (hf : Continuous f) (h'f : HasCompactSupport f) :
    StronglyMeasurable f 
goal: MeasureTheory.StronglyMeasurable f

full_name: Embedding.comp_stronglyMeasurable_iff
code: /-- If `g` is a topological embedding, then `f` is strongly measurable iff `g ∘ f` is. -/
theorem _root_.Embedding.comp_stronglyMeasurable_iff {m : MeasurableSpace α} [TopologicalSpace β]
    [PseudoMetrizableSpace β] [TopologicalSpace γ] [PseudoMetrizableSpace γ] {g : β → γ} {f : α → β}
    (hg : Embedding g) : (StronglyMeasurable fun x => g (f x)) ↔ StronglyMeasurable f 
goal: (MeasureTheory.StronglyMeasurable fun x => g (f x)) ↔ MeasureTheory.StronglyMeasurable f

full_name: stronglyMeasurable_of_tendsto
code: /-- A sequential limit of strongly measurable functions is strongly measurable. -/
theorem _root_.stronglyMeasurable_of_tendsto {ι : Type*} {m : MeasurableSpace α}
    [TopologicalSpace β] [PseudoMetrizableSpace β] (u : Filter ι) [NeBot u] [IsCountablyGenerated u]
    {f : ι → α → β} {g : α → β} (hf : ∀ i, StronglyMeasurable (f i)) (lim : Tendsto f u (𝓝 g)) :
    StronglyMeasurable g 
goal: MeasureTheory.StronglyMeasurable g

full_name: MeasureTheory.StronglyMeasurable.piecewise
code: protected theorem piecewise {m : MeasurableSpace α} [TopologicalSpace β] {s : Set α}
    {_ : DecidablePred (· ∈ s)} (hs : MeasurableSet s) (hf : StronglyMeasurable f)
    (hg : StronglyMeasurable g) : StronglyMeasurable (Set.piecewise s f g) 
goal: ∀ {x : DecidablePred fun x => x ∈ s},
  MeasurableSet s →
    MeasureTheory.StronglyMeasurable f →
      MeasureTheory.StronglyMeasurable g → MeasureTheory.StronglyMeasurable (s.piecewise f g)

full_name: MeasureTheory.StronglyMeasurable.ite
code: /-- this is slightly different from `StronglyMeasurable.piecewise`. It can be used to show
`StronglyMeasurable (ite (x=0) 0 1)` by
`exact StronglyMeasurable.ite (measurableSet_singleton 0) stronglyMeasurable_const
stronglyMeasurable_const`, but replacing `StronglyMeasurable.ite` by
`StronglyMeasurable.piecewise` in that example proof does not work. -/
protected theorem ite {_ : MeasurableSpace α} [TopologicalSpace β] {p : α → Prop}
    {_ : DecidablePred p} (hp : MeasurableSet { a : α | p a }) (hf : StronglyMeasurable f)
    (hg : StronglyMeasurable g) : StronglyMeasurable fun x => ite (p x) (f x) (g x) 
goal: ∀ {x : MeasurableSpace α} [inst : TopologicalSpace β] {p : α → Prop} {x_1 : DecidablePred p},
  MeasurableSet {a | p a} →
    MeasureTheory.StronglyMeasurable f →
      MeasureTheory.StronglyMeasurable g → MeasureTheory.StronglyMeasurable fun x => if p x then f x else g x

full_name: MeasurableEmbedding.stronglyMeasurable_extend
code: @[measurability]
theorem _root_.MeasurableEmbedding.stronglyMeasurable_extend {f : α → β} {g : α → γ} {g' : γ → β}
    {mα : MeasurableSpace α} {mγ : MeasurableSpace γ} [TopologicalSpace β]
    (hg : MeasurableEmbedding g) (hf : StronglyMeasurable f) (hg' : StronglyMeasurable g') :
    StronglyMeasurable (Function.extend g f g') 
goal: MeasureTheory.StronglyMeasurable (Function.extend g f g')

full_name: MeasurableEmbedding.exists_stronglyMeasurable_extend
code: theorem _root_.MeasurableEmbedding.exists_stronglyMeasurable_extend {f : α → β} {g : α → γ}
    {_ : MeasurableSpace α} {_ : MeasurableSpace γ} [TopologicalSpace β]
    (hg : MeasurableEmbedding g) (hf : StronglyMeasurable f) (hne : γ → Nonempty β) :
    ∃ f' : γ → β, StronglyMeasurable f' ∧ f' ∘ g = f 
goal: ∀ {x : MeasurableSpace α} {x_1 : MeasurableSpace γ} [inst : TopologicalSpace β],
  MeasurableEmbedding g →
    MeasureTheory.StronglyMeasurable f → (γ → Nonempty β) → ∃ f', MeasureTheory.StronglyMeasurable f' ∧ f' ∘ g = f

full_name: stronglyMeasurable_of_stronglyMeasurable_union_cover
code: theorem _root_.stronglyMeasurable_of_stronglyMeasurable_union_cover {m : MeasurableSpace α}
    [TopologicalSpace β] {f : α → β} (s t : Set α) (hs : MeasurableSet s) (ht : MeasurableSet t)
    (h : univ ⊆ s ∪ t) (hc : StronglyMeasurable fun a : s => f a)
    (hd : StronglyMeasurable fun a : t => f a) : StronglyMeasurable f 
goal: MeasureTheory.StronglyMeasurable f

full_name: stronglyMeasurable_of_restrict_of_restrict_compl
code: theorem _root_.stronglyMeasurable_of_restrict_of_restrict_compl {_ : MeasurableSpace α}
    [TopologicalSpace β] {f : α → β} {s : Set α} (hs : MeasurableSet s)
    (h₁ : StronglyMeasurable (s.restrict f)) (h₂ : StronglyMeasurable (sᶜ.restrict f)) :
    StronglyMeasurable f 
goal: ∀ {x : MeasurableSpace α} [inst : TopologicalSpace β] {f : α → β} {s : Set α},
  MeasurableSet s →
    MeasureTheory.StronglyMeasurable (s.restrict f) →
      MeasureTheory.StronglyMeasurable (sᶜ.restrict f) → MeasureTheory.StronglyMeasurable f

full_name: MeasureTheory.StronglyMeasurable.indicator
code: @[measurability]
protected theorem indicator {_ : MeasurableSpace α} [TopologicalSpace β] [Zero β]
    (hf : StronglyMeasurable f) {s : Set α} (hs : MeasurableSet s) :
    StronglyMeasurable (s.indicator f) 
goal: ∀ {x : MeasurableSpace α} [inst : TopologicalSpace β] [inst_1 : Zero β],
  MeasureTheory.StronglyMeasurable f → ∀ {s : Set α}, MeasurableSet s → MeasureTheory.StronglyMeasurable (s.indicator f)

full_name: MeasureTheory.StronglyMeasurable.dist
code: @[aesop safe 20 apply (rule_sets := [Measurable])]
protected theorem dist {_ : MeasurableSpace α} {β : Type*} [PseudoMetricSpace β] {f g : α → β}
    (hf : StronglyMeasurable f) (hg : StronglyMeasurable g) :
    StronglyMeasurable fun x => dist (f x) (g x) 
goal: ∀ {x : MeasurableSpace α} {β : Type u_5} [inst : PseudoMetricSpace β] {f g : α → β},
  MeasureTheory.StronglyMeasurable f →
    MeasureTheory.StronglyMeasurable g → MeasureTheory.StronglyMeasurable fun x => dist (f x) (g x)

full_name: MeasureTheory.StronglyMeasurable.norm
code: @[measurability]
protected theorem norm {_ : MeasurableSpace α} {β : Type*} [SeminormedAddCommGroup β] {f : α → β}
    (hf : StronglyMeasurable f) : StronglyMeasurable fun x => ‖f x‖ 
goal: ∀ {x : MeasurableSpace α} {β : Type u_5} [inst : SeminormedAddCommGroup β] {f : α → β},
  MeasureTheory.StronglyMeasurable f → MeasureTheory.StronglyMeasurable fun x => ‖f x‖

full_name: MeasureTheory.StronglyMeasurable.nnnorm
code: @[measurability]
protected theorem nnnorm {_ : MeasurableSpace α} {β : Type*} [SeminormedAddCommGroup β] {f : α → β}
    (hf : StronglyMeasurable f) : StronglyMeasurable fun x => ‖f x‖₊ 
goal: ∀ {x : MeasurableSpace α} {β : Type u_5} [inst : SeminormedAddCommGroup β] {f : α → β},
  MeasureTheory.StronglyMeasurable f → MeasureTheory.StronglyMeasurable fun x => ‖f x‖₊

full_name: MeasureTheory.StronglyMeasurable.ennnorm
code: @[measurability]
protected theorem ennnorm {_ : MeasurableSpace α} {β : Type*} [SeminormedAddCommGroup β]
    {f : α → β} (hf : StronglyMeasurable f) : Measurable fun a => (‖f a‖₊ : ℝ≥0∞) 
goal: ∀ {x : MeasurableSpace α} {β : Type u_5} [inst : SeminormedAddCommGroup β] {f : α → β},
  MeasureTheory.StronglyMeasurable f → Measurable fun a => ↑‖f a‖₊

full_name: MeasureTheory.StronglyMeasurable.real_toNNReal
code: @[measurability]
protected theorem real_toNNReal {_ : MeasurableSpace α} {f : α → ℝ} (hf : StronglyMeasurable f) :
    StronglyMeasurable fun x => (f x).toNNReal 
goal: ∀ {x : MeasurableSpace α} {f : α → ℝ},
  MeasureTheory.StronglyMeasurable f → MeasureTheory.StronglyMeasurable fun x => (f x).toNNReal

full_name: MeasureTheory.StronglyMeasurable.measurableSet_eq_fun
code: theorem measurableSet_eq_fun {m : MeasurableSpace α} {E} [TopologicalSpace E] [MetrizableSpace E]
    {f g : α → E} (hf : StronglyMeasurable f) (hg : StronglyMeasurable g) :
    MeasurableSet { x | f x = g x } 
goal: MeasurableSet {x | f x = g x}

full_name: MeasureTheory.StronglyMeasurable.measurableSet_lt
code: theorem measurableSet_lt {m : MeasurableSpace α} [TopologicalSpace β] [LinearOrder β]
    [OrderClosedTopology β] [PseudoMetrizableSpace β] {f g : α → β} (hf : StronglyMeasurable f)
    (hg : StronglyMeasurable g) : MeasurableSet { a | f a < g a } 
goal: MeasurableSet {a | f a < g a}

full_name: MeasureTheory.StronglyMeasurable.measurableSet_le
code: theorem measurableSet_le {m : MeasurableSpace α} [TopologicalSpace β] [Preorder β]
    [OrderClosedTopology β] [PseudoMetrizableSpace β] {f g : α → β} (hf : StronglyMeasurable f)
    (hg : StronglyMeasurable g) : MeasurableSet { a | f a ≤ g a } 
goal: MeasurableSet {a | f a ≤ g a}

full_name: MeasureTheory.StronglyMeasurable.stronglyMeasurable_in_set
code: theorem stronglyMeasurable_in_set {m : MeasurableSpace α} [TopologicalSpace β] [Zero β] {s : Set α}
    {f : α → β} (hs : MeasurableSet s) (hf : StronglyMeasurable f)
    (hf_zero : ∀ x, x ∉ s → f x = 0) :
    ∃ fs : ℕ → α →ₛ β,
      (∀ x, Tendsto (fun n => fs n x) atTop (𝓝 (f x))) ∧ ∀ x ∉ s, ∀ n, fs n x = 0 
goal: ∃ fs, (∀ (x : α), Filter.Tendsto (fun n => ↑(fs n) x) Filter.atTop (𝓝 (f x))) ∧ ∀ x ∉ s, ∀ (n : ℕ), ↑(fs n) x = 0

full_name: MeasureTheory.StronglyMeasurable.stronglyMeasurable_of_measurableSpace_le_on
code: /-- If the restriction to a set `s` of a σ-algebra `m` is included in the restriction to `s` of
another σ-algebra `m₂` (hypothesis `hs`), the set `s` is `m` measurable and a function `f` supported
on `s` is `m`-strongly-measurable, then `f` is also `m₂`-strongly-measurable. -/
theorem stronglyMeasurable_of_measurableSpace_le_on {α E} {m m₂ : MeasurableSpace α}
    [TopologicalSpace E] [Zero E] {s : Set α} {f : α → E} (hs_m : MeasurableSet[m] s)
    (hs : ∀ t, MeasurableSet[m] (s ∩ t) → MeasurableSet[m₂] (s ∩ t))
    (hf : StronglyMeasurable[m] f) (hf_zero : ∀ x ∉ s, f x = 0) :
    StronglyMeasurable[m₂] f 
goal: MeasureTheory.StronglyMeasurable f

full_name: MeasureTheory.StronglyMeasurable.exists_spanning_measurableSet_norm_le
code: /-- If a function `f` is strongly measurable w.r.t. a sub-σ-algebra `m` and the measure is σ-finite
on `m`, then there exists spanning measurable sets with finite measure on which `f` has bounded
norm. In particular, `f` is integrable on each of those sets. -/
theorem exists_spanning_measurableSet_norm_le [SeminormedAddCommGroup β] {m m0 : MeasurableSpace α}
    (hm : m ≤ m0) (hf : StronglyMeasurable[m] f) (μ : Measure α) [SigmaFinite (μ.trim hm)] :
    ∃ s : ℕ → Set α,
      (∀ n, MeasurableSet[m] (s n) ∧ μ (s n) < ∞ ∧ ∀ x ∈ s n, ‖f x‖ ≤ n) ∧
      ⋃ i, s i = Set.univ 
goal: ∃ s, (∀ (n : ℕ), MeasurableSet (s n) ∧ μ (s n) < ⊤ ∧ ∀ x ∈ s n, ‖f x‖ ≤ ↑n) ∧ ⋃ i, s i = Set.univ

full_name: MeasureTheory.finStronglyMeasurable_zero
code: theorem finStronglyMeasurable_zero {α β} {m : MeasurableSpace α} {μ : Measure α} [Zero β]
    [TopologicalSpace β] : FinStronglyMeasurable (0 : α → β) μ 
goal: MeasureTheory.FinStronglyMeasurable 0 μ

full_name: MeasureTheory.FinStronglyMeasurable.aefinStronglyMeasurable
code: theorem aefinStronglyMeasurable [Zero β] [TopologicalSpace β] (hf : FinStronglyMeasurable f μ) :
    AEFinStronglyMeasurable f μ 
goal: MeasureTheory.AEFinStronglyMeasurable f μ

full_name: MeasureTheory.FinStronglyMeasurable.approx
code: /-- A sequence of simple functions such that `∀ x, Tendsto (fun n ↦ hf.approx n x) atTop (𝓝 (f x))`
and `∀ n, μ (support (hf.approx n)) < ∞`. These properties are given by
`FinStronglyMeasurable.tendsto_approx` and `FinStronglyMeasurable.fin_support_approx`. -/
protected noncomputable def approx : ℕ → α →ₛ β 
goal: ℕ → MeasureTheory.SimpleFunc α β

full_name: MeasureTheory.FinStronglyMeasurable.fin_support_approx
code: protected theorem fin_support_approx : ∀ n, μ (support (hf.approx n)) < ∞ 
goal: μ (Function.support ↑(hf.approx n)) < ⊤

full_name: MeasureTheory.FinStronglyMeasurable.tendsto_approx
code: protected theorem tendsto_approx : ∀ x, Tendsto (fun n => hf.approx n x) atTop (𝓝 (f x)) 
goal: Filter.Tendsto (fun n => ↑(hf.approx n) x) Filter.atTop (𝓝 (f x))

full_name: MeasureTheory.FinStronglyMeasurable.stronglyMeasurable
code: /-- A finitely strongly measurable function is strongly measurable. -/
@[aesop 5% apply (rule_sets := [Measurable])]
protected theorem stronglyMeasurable [Zero β] [TopologicalSpace β]
    (hf : FinStronglyMeasurable f μ) : StronglyMeasurable f 
goal: MeasureTheory.StronglyMeasurable f

full_name: MeasureTheory.FinStronglyMeasurable.exists_set_sigmaFinite
code: theorem exists_set_sigmaFinite [Zero β] [TopologicalSpace β] [T2Space β]
    (hf : FinStronglyMeasurable f μ) :
    ∃ t, MeasurableSet t ∧ (∀ x ∈ tᶜ, f x = 0) ∧ SigmaFinite (μ.restrict t) 
goal: ∃ t, MeasurableSet t ∧ (∀ x ∈ tᶜ, f x = 0) ∧ MeasureTheory.SigmaFinite (μ.restrict t)

full_name: MeasureTheory.FinStronglyMeasurable.measurable
code: /-- A finitely strongly measurable function is measurable. -/
protected theorem measurable [Zero β] [TopologicalSpace β] [PseudoMetrizableSpace β]
    [MeasurableSpace β] [BorelSpace β] (hf : FinStronglyMeasurable f μ) : Measurable f 
goal: Measurable f

full_name: MeasureTheory.FinStronglyMeasurable.mul
code: @[aesop safe 20 (rule_sets := [Measurable])]
protected theorem mul [MonoidWithZero β] [ContinuousMul β] (hf : FinStronglyMeasurable f μ)
    (hg : FinStronglyMeasurable g μ) : FinStronglyMeasurable (f * g) μ 
goal: MeasureTheory.FinStronglyMeasurable (f * g) μ

full_name: MeasureTheory.FinStronglyMeasurable.add
code: @[aesop safe 20 (rule_sets := [Measurable])]
protected theorem add [AddMonoid β] [ContinuousAdd β] (hf : FinStronglyMeasurable f μ)
    (hg : FinStronglyMeasurable g μ) : FinStronglyMeasurable (f + g) μ 
goal: MeasureTheory.FinStronglyMeasurable (f + g) μ

full_name: MeasureTheory.FinStronglyMeasurable.neg
code: @[measurability]
protected theorem neg [AddGroup β] [TopologicalAddGroup β] (hf : FinStronglyMeasurable f μ) :
    FinStronglyMeasurable (-f) μ 
goal: MeasureTheory.FinStronglyMeasurable (-f) μ

full_name: MeasureTheory.FinStronglyMeasurable.sub
code: @[measurability]
protected theorem sub [AddGroup β] [ContinuousSub β] (hf : FinStronglyMeasurable f μ)
    (hg : FinStronglyMeasurable g μ) : FinStronglyMeasurable (f - g) μ 
goal: MeasureTheory.FinStronglyMeasurable (f - g) μ

full_name: MeasureTheory.FinStronglyMeasurable.const_smul
code: @[measurability]
protected theorem const_smul {𝕜} [TopologicalSpace 𝕜] [AddMonoid β] [Monoid 𝕜]
    [DistribMulAction 𝕜 β] [ContinuousSMul 𝕜 β] (hf : FinStronglyMeasurable f μ) (c : 𝕜) :
    FinStronglyMeasurable (c • f) μ 
goal: MeasureTheory.FinStronglyMeasurable (c • f) μ

full_name: MeasureTheory.FinStronglyMeasurable.sup
code: @[aesop safe 20 (rule_sets := [Measurable])]
protected theorem sup [SemilatticeSup β] [ContinuousSup β] (hf : FinStronglyMeasurable f μ)
    (hg : FinStronglyMeasurable g μ) : FinStronglyMeasurable (f ⊔ g) μ 
goal: MeasureTheory.FinStronglyMeasurable (f ⊔ g) μ

full_name: MeasureTheory.FinStronglyMeasurable.inf
code: @[aesop safe 20 (rule_sets := [Measurable])]
protected theorem inf [SemilatticeInf β] [ContinuousInf β] (hf : FinStronglyMeasurable f μ)
    (hg : FinStronglyMeasurable g μ) : FinStronglyMeasurable (f ⊓ g) μ 
goal: MeasureTheory.FinStronglyMeasurable (f ⊓ g) μ

full_name: MeasureTheory.finStronglyMeasurable_iff_stronglyMeasurable_and_exists_set_sigmaFinite
code: theorem finStronglyMeasurable_iff_stronglyMeasurable_and_exists_set_sigmaFinite {α β} {f : α → β}
    [TopologicalSpace β] [T2Space β] [Zero β] {_ : MeasurableSpace α} {μ : Measure α} :
    FinStronglyMeasurable f μ ↔
      StronglyMeasurable f ∧
        ∃ t, MeasurableSet t ∧ (∀ x ∈ tᶜ, f x = 0) ∧ SigmaFinite (μ.restrict t) 
goal: ∀ {x : MeasurableSpace α} {μ : MeasureTheory.Measure α},
  MeasureTheory.FinStronglyMeasurable f μ ↔
    MeasureTheory.StronglyMeasurable f ∧
      ∃ t, MeasurableSet t ∧ (∀ x ∈ tᶜ, f x = 0) ∧ MeasureTheory.SigmaFinite (μ.restrict t)

full_name: MeasureTheory.aefinStronglyMeasurable_zero
code: theorem aefinStronglyMeasurable_zero {α β} {_ : MeasurableSpace α} (μ : Measure α) [Zero β]
    [TopologicalSpace β] : AEFinStronglyMeasurable (0 : α → β) μ 
goal: ∀ {x : MeasurableSpace α} (μ : MeasureTheory.Measure α) [inst : Zero β] [inst_1 : TopologicalSpace β],
  MeasureTheory.AEFinStronglyMeasurable 0 μ

full_name: MeasureTheory.aestronglyMeasurable_const
code: @[measurability]
theorem aestronglyMeasurable_const {α β} {_ : MeasurableSpace α} {μ : Measure α}
    [TopologicalSpace β] {b : β} : AEStronglyMeasurable (fun _ : α => b) μ 
goal: ∀ {x : MeasurableSpace α} {μ : MeasureTheory.Measure α} [inst : TopologicalSpace β] {b : β},
  MeasureTheory.AEStronglyMeasurable (fun x => b) μ

full_name: MeasureTheory.aestronglyMeasurable_zero
code: @[to_additive (attr := measurability)]
theorem aestronglyMeasurable_one {α β} {_ : MeasurableSpace α} {μ : Measure α} [TopologicalSpace β]
    [One β] : AEStronglyMeasurable (1 : α → β) μ 
goal: ∀ {x : MeasurableSpace α} {μ : MeasureTheory.Measure α} [inst : TopologicalSpace β] [inst_1 : Zero β],
  MeasureTheory.AEStronglyMeasurable 0 μ

full_name: MeasureTheory.aestronglyMeasurable_one
code: @[to_additive (attr := measurability)]
theorem aestronglyMeasurable_one {α β} {_ : MeasurableSpace α} {μ : Measure α} [TopologicalSpace β]
    [One β] : AEStronglyMeasurable (1 : α → β) μ 
goal: ∀ {x : MeasurableSpace α} {μ : MeasureTheory.Measure α} [inst : TopologicalSpace β] [inst_1 : One β],
  MeasureTheory.AEStronglyMeasurable 1 μ

full_name: MeasureTheory.Subsingleton.aestronglyMeasurable
code: @[simp]
theorem Subsingleton.aestronglyMeasurable {_ : MeasurableSpace α} [TopologicalSpace β]
    [Subsingleton β] {μ : Measure α} (f : α → β) : AEStronglyMeasurable f μ 
goal: ∀ {x : MeasurableSpace α} [inst : TopologicalSpace β] [inst_1 : Subsingleton β] {μ : MeasureTheory.Measure α}
  (f : α → β), MeasureTheory.AEStronglyMeasurable f μ

full_name: MeasureTheory.Subsingleton.aestronglyMeasurable'
code: @[simp]
theorem Subsingleton.aestronglyMeasurable' {_ : MeasurableSpace α} [TopologicalSpace β]
    [Subsingleton α] {μ : Measure α} (f : α → β) : AEStronglyMeasurable f μ 
goal: ∀ {x : MeasurableSpace α} [inst : TopologicalSpace β] [inst_1 : Subsingleton α] {μ : MeasureTheory.Measure α}
  (f : α → β), MeasureTheory.AEStronglyMeasurable f μ

full_name: MeasureTheory.aestronglyMeasurable_zero_measure
code: @[simp]
theorem aestronglyMeasurable_zero_measure [MeasurableSpace α] [TopologicalSpace β] (f : α → β) :
    AEStronglyMeasurable f (0 : Measure α) 
goal: MeasureTheory.AEStronglyMeasurable f 0

full_name: MeasureTheory.SimpleFunc.aestronglyMeasurable
code: @[measurability]
theorem SimpleFunc.aestronglyMeasurable {_ : MeasurableSpace α} {μ : Measure α} [TopologicalSpace β]
    (f : α →ₛ β) : AEStronglyMeasurable f μ 
goal: ∀ {x : MeasurableSpace α} {μ : MeasureTheory.Measure α} [inst : TopologicalSpace β] (f : MeasureTheory.SimpleFunc α β),
  MeasureTheory.AEStronglyMeasurable (↑f) μ

full_name: MeasureTheory.AEStronglyMeasurable.mk
code: /-- A `StronglyMeasurable` function such that `f =ᵐ[μ] hf.mk f`. See lemmas
`stronglyMeasurable_mk` and `ae_eq_mk`. -/
protected noncomputable def mk (f : α → β) (hf : AEStronglyMeasurable f μ) : α → β 
goal: α → β

full_name: MeasureTheory.AEStronglyMeasurable.stronglyMeasurable_mk
code: theorem stronglyMeasurable_mk (hf : AEStronglyMeasurable f μ) : StronglyMeasurable (hf.mk f) 
goal: MeasureTheory.StronglyMeasurable (MeasureTheory.AEStronglyMeasurable.mk f hf)

full_name: MeasureTheory.AEStronglyMeasurable.measurable_mk
code: theorem measurable_mk [PseudoMetrizableSpace β] [MeasurableSpace β] [BorelSpace β]
    (hf : AEStronglyMeasurable f μ) : Measurable (hf.mk f) 
goal: Measurable (MeasureTheory.AEStronglyMeasurable.mk f hf)

full_name: MeasureTheory.AEStronglyMeasurable.ae_eq_mk
code: theorem ae_eq_mk (hf : AEStronglyMeasurable f μ) : f =ᵐ[μ] hf.mk f 
goal: f =ᵐ[μ] MeasureTheory.AEStronglyMeasurable.mk f hf

full_name: MeasureTheory.AEStronglyMeasurable.aemeasurable
code: @[aesop 5% apply (rule_sets := [Measurable])]
protected theorem aemeasurable {β} [MeasurableSpace β] [TopologicalSpace β]
    [PseudoMetrizableSpace β] [BorelSpace β] {f : α → β} (hf : AEStronglyMeasurable f μ) :
    AEMeasurable f μ 
goal: AEMeasurable f μ

full_name: MeasureTheory.AEStronglyMeasurable.congr
code: theorem congr (hf : AEStronglyMeasurable f μ) (h : f =ᵐ[μ] g) : AEStronglyMeasurable g μ 
goal: MeasureTheory.AEStronglyMeasurable g μ

full_name: aestronglyMeasurable_congr
code: theorem _root_.aestronglyMeasurable_congr (h : f =ᵐ[μ] g) :
    AEStronglyMeasurable f μ ↔ AEStronglyMeasurable g μ 
goal: MeasureTheory.AEStronglyMeasurable f μ ↔ MeasureTheory.AEStronglyMeasurable g μ

full_name: MeasureTheory.AEStronglyMeasurable.mono_measure
code: theorem mono_measure {ν : Measure α} (hf : AEStronglyMeasurable f μ) (h : ν ≤ μ) :
    AEStronglyMeasurable f ν 
goal: MeasureTheory.AEStronglyMeasurable f ν

full_name: MeasureTheory.AEStronglyMeasurable.mono_ac
code: protected lemma mono_ac (h : ν ≪ μ) (hμ : AEStronglyMeasurable f μ) : AEStronglyMeasurable f ν 
goal: MeasureTheory.AEStronglyMeasurable f ν

full_name: MeasureTheory.AEStronglyMeasurable.mono_set
code: theorem mono_set {s t} (h : s ⊆ t) (ht : AEStronglyMeasurable f (μ.restrict t)) :
    AEStronglyMeasurable f (μ.restrict s) 
goal: MeasureTheory.AEStronglyMeasurable f (μ.restrict s)

full_name: MeasureTheory.AEStronglyMeasurable.restrict
code: protected theorem restrict (hfm : AEStronglyMeasurable f μ) {s} :
    AEStronglyMeasurable f (μ.restrict s) 
goal: MeasureTheory.AEStronglyMeasurable f (μ.restrict s)

full_name: MeasureTheory.AEStronglyMeasurable.ae_mem_imp_eq_mk
code: theorem ae_mem_imp_eq_mk {s} (h : AEStronglyMeasurable f (μ.restrict s)) :
    ∀ᵐ x ∂μ, x ∈ s → f x = h.mk f x 
goal: ∀ᵐ (x : α) ∂μ, x ∈ s → f x = MeasureTheory.AEStronglyMeasurable.mk f h x

full_name: Continuous.comp_aestronglyMeasurable
code: /-- The composition of a continuous function and an ae strongly measurable function is ae strongly
measurable. -/
theorem _root_.Continuous.comp_aestronglyMeasurable {g : β → γ} {f : α → β} (hg : Continuous g)
    (hf : AEStronglyMeasurable f μ) : AEStronglyMeasurable (fun x => g (f x)) μ 
goal: MeasureTheory.AEStronglyMeasurable (fun x => g (f x)) μ

full_name: Continuous.aestronglyMeasurable
code: /-- A continuous function from `α` to `β` is ae strongly measurable when one of the two spaces is
second countable. -/
theorem _root_.Continuous.aestronglyMeasurable [TopologicalSpace α] [OpensMeasurableSpace α]
    [PseudoMetrizableSpace β] [SecondCountableTopologyEither α β] (hf : Continuous f) :
    AEStronglyMeasurable f μ 
goal: MeasureTheory.AEStronglyMeasurable f μ

full_name: MeasureTheory.AEStronglyMeasurable.prod_mk
code: protected theorem prod_mk {f : α → β} {g : α → γ} (hf : AEStronglyMeasurable f μ)
    (hg : AEStronglyMeasurable g μ) : AEStronglyMeasurable (fun x => (f x, g x)) μ 
goal: MeasureTheory.AEStronglyMeasurable (fun x => (f x, g x)) μ

full_name: Continuous.comp_aestronglyMeasurable₂
code: /-- The composition of a continuous function of two variables and two ae strongly measurable
functions is ae strongly measurable. -/
theorem _root_.Continuous.comp_aestronglyMeasurable₂
    {β' : Type*} [TopologicalSpace β']
    {g : β → β' → γ} {f : α → β} {f' : α → β'} (hg : Continuous g.uncurry)
    (hf : AEStronglyMeasurable f μ) (h'f : AEStronglyMeasurable f' μ) :
    AEStronglyMeasurable (fun x => g (f x) (f' x)) μ 
goal: MeasureTheory.AEStronglyMeasurable (fun x => g (f x) (f' x)) μ

full_name: Measurable.aestronglyMeasurable
code: /-- In a space with second countable topology, measurable implies ae strongly measurable. -/
@[fun_prop, aesop unsafe 30% apply (rule_sets := [Measurable])]
theorem _root_.Measurable.aestronglyMeasurable {_ : MeasurableSpace α} {μ : Measure α}
    [MeasurableSpace β] [PseudoMetrizableSpace β] [SecondCountableTopology β]
    [OpensMeasurableSpace β] (hf : Measurable f) : AEStronglyMeasurable f μ 
goal: ∀ {x : MeasurableSpace α} {μ : MeasureTheory.Measure α} [inst : MeasurableSpace β]
  [inst_1 : TopologicalSpace.PseudoMetrizableSpace β] [inst_2 : SecondCountableTopology β]
  [inst_3 : OpensMeasurableSpace β], Measurable f → MeasureTheory.AEStronglyMeasurable f μ

full_name: MeasureTheory.AEStronglyMeasurable.mul
code: @[to_additive (attr := aesop safe 20 apply (rule_sets := [Measurable]))]
protected theorem mul [Mul β] [ContinuousMul β] (hf : AEStronglyMeasurable f μ)
    (hg : AEStronglyMeasurable g μ) : AEStronglyMeasurable (f * g) μ 
goal: MeasureTheory.AEStronglyMeasurable (f * g) μ

full_name: MeasureTheory.AEStronglyMeasurable.add
code: @[to_additive (attr := aesop safe 20 apply (rule_sets := [Measurable]))]
protected theorem mul [Mul β] [ContinuousMul β] (hf : AEStronglyMeasurable f μ)
    (hg : AEStronglyMeasurable g μ) : AEStronglyMeasurable (f * g) μ 
goal: MeasureTheory.AEStronglyMeasurable (f + g) μ

full_name: MeasureTheory.AEStronglyMeasurable.add_const
code: @[to_additive (attr := measurability)]
protected theorem mul_const [Mul β] [ContinuousMul β] (hf : AEStronglyMeasurable f μ) (c : β) :
    AEStronglyMeasurable (fun x => f x * c) μ 
goal: MeasureTheory.AEStronglyMeasurable (fun x => f x + c) μ

full_name: MeasureTheory.AEStronglyMeasurable.mul_const
code: @[to_additive (attr := measurability)]
protected theorem mul_const [Mul β] [ContinuousMul β] (hf : AEStronglyMeasurable f μ) (c : β) :
    AEStronglyMeasurable (fun x => f x * c) μ 
goal: MeasureTheory.AEStronglyMeasurable (fun x => f x * c) μ

full_name: MeasureTheory.AEStronglyMeasurable.const_add
code: @[to_additive (attr := measurability)]
protected theorem const_mul [Mul β] [ContinuousMul β] (hf : AEStronglyMeasurable f μ) (c : β) :
    AEStronglyMeasurable (fun x => c * f x) μ 
goal: MeasureTheory.AEStronglyMeasurable (fun x => c + f x) μ

full_name: MeasureTheory.AEStronglyMeasurable.const_mul
code: @[to_additive (attr := measurability)]
protected theorem const_mul [Mul β] [ContinuousMul β] (hf : AEStronglyMeasurable f μ) (c : β) :
    AEStronglyMeasurable (fun x => c * f x) μ 
goal: MeasureTheory.AEStronglyMeasurable (fun x => c * f x) μ

full_name: MeasureTheory.AEStronglyMeasurable.inv
code: @[to_additive (attr := measurability)]
protected theorem inv [Inv β] [ContinuousInv β] (hf : AEStronglyMeasurable f μ) :
    AEStronglyMeasurable f⁻¹ μ 
goal: MeasureTheory.AEStronglyMeasurable f⁻¹ μ

full_name: MeasureTheory.AEStronglyMeasurable.neg
code: @[to_additive (attr := measurability)]
protected theorem inv [Inv β] [ContinuousInv β] (hf : AEStronglyMeasurable f μ) :
    AEStronglyMeasurable f⁻¹ μ 
goal: MeasureTheory.AEStronglyMeasurable (-f) μ

full_name: MeasureTheory.AEStronglyMeasurable.sub
code: @[to_additive (attr := aesop safe 20 apply (rule_sets := [Measurable]))]
protected theorem div [Group β] [TopologicalGroup β] (hf : AEStronglyMeasurable f μ)
    (hg : AEStronglyMeasurable g μ) : AEStronglyMeasurable (f / g) μ 
goal: MeasureTheory.AEStronglyMeasurable (f - g) μ

full_name: MeasureTheory.AEStronglyMeasurable.div
code: @[to_additive (attr := aesop safe 20 apply (rule_sets := [Measurable]))]
protected theorem div [Group β] [TopologicalGroup β] (hf : AEStronglyMeasurable f μ)
    (hg : AEStronglyMeasurable g μ) : AEStronglyMeasurable (f / g) μ 
goal: MeasureTheory.AEStronglyMeasurable (f / g) μ

full_name: MeasureTheory.AEStronglyMeasurable.mul_iff_right
code: @[to_additive]
theorem mul_iff_right [CommGroup β] [TopologicalGroup β] (hf : AEStronglyMeasurable f μ) :
    AEStronglyMeasurable (f * g) μ ↔ AEStronglyMeasurable g μ 
goal: MeasureTheory.AEStronglyMeasurable (f * g) μ ↔ MeasureTheory.AEStronglyMeasurable g μ

full_name: MeasureTheory.AEStronglyMeasurable.add_iff_right
code: @[to_additive]
theorem mul_iff_right [CommGroup β] [TopologicalGroup β] (hf : AEStronglyMeasurable f μ) :
    AEStronglyMeasurable (f * g) μ ↔ AEStronglyMeasurable g μ 
goal: MeasureTheory.AEStronglyMeasurable (f + g) μ ↔ MeasureTheory.AEStronglyMeasurable g μ

full_name: MeasureTheory.AEStronglyMeasurable.mul_iff_left
code: @[to_additive]
theorem mul_iff_left [CommGroup β] [TopologicalGroup β] (hf : AEStronglyMeasurable f μ) :
    AEStronglyMeasurable (g * f) μ ↔ AEStronglyMeasurable g μ 
goal: MeasureTheory.AEStronglyMeasurable (g * f) μ ↔ MeasureTheory.AEStronglyMeasurable g μ

full_name: MeasureTheory.AEStronglyMeasurable.add_iff_left
code: @[to_additive]
theorem mul_iff_left [CommGroup β] [TopologicalGroup β] (hf : AEStronglyMeasurable f μ) :
    AEStronglyMeasurable (g * f) μ ↔ AEStronglyMeasurable g μ 
goal: MeasureTheory.AEStronglyMeasurable (g + f) μ ↔ MeasureTheory.AEStronglyMeasurable g μ

full_name: MeasureTheory.AEStronglyMeasurable.smul
code: @[to_additive (attr := aesop safe 20 apply (rule_sets := [Measurable]))]
protected theorem smul {𝕜} [TopologicalSpace 𝕜] [SMul 𝕜 β] [ContinuousSMul 𝕜 β] {f : α → 𝕜}
    {g : α → β} (hf : AEStronglyMeasurable f μ) (hg : AEStronglyMeasurable g μ) :
    AEStronglyMeasurable (fun x => f x • g x) μ 
goal: MeasureTheory.AEStronglyMeasurable (fun x => f x • g x) μ

full_name: MeasureTheory.AEStronglyMeasurable.vadd
code: @[to_additive (attr := aesop safe 20 apply (rule_sets := [Measurable]))]
protected theorem smul {𝕜} [TopologicalSpace 𝕜] [SMul 𝕜 β] [ContinuousSMul 𝕜 β] {f : α → 𝕜}
    {g : α → β} (hf : AEStronglyMeasurable f μ) (hg : AEStronglyMeasurable g μ) :
    AEStronglyMeasurable (fun x => f x • g x) μ 
goal: MeasureTheory.AEStronglyMeasurable (fun x => f x +ᵥ g x) μ

full_name: MeasureTheory.AEStronglyMeasurable.pow
code: @[to_additive (attr := aesop safe 20 apply (rule_sets := [Measurable])) const_nsmul]
protected theorem pow [Monoid β] [ContinuousMul β] (hf : AEStronglyMeasurable f μ) (n : ℕ) :
    AEStronglyMeasurable (f ^ n) μ 
goal: MeasureTheory.AEStronglyMeasurable (f ^ n) μ

full_name: MeasureTheory.AEStronglyMeasurable.const_nsmul
code: @[to_additive (attr := aesop safe 20 apply (rule_sets := [Measurable])) const_nsmul]
protected theorem pow [Monoid β] [ContinuousMul β] (hf : AEStronglyMeasurable f μ) (n : ℕ) :
    AEStronglyMeasurable (f ^ n) μ 
goal: MeasureTheory.AEStronglyMeasurable (n • f) μ

full_name: MeasureTheory.AEStronglyMeasurable.const_vadd
code: @[to_additive (attr := measurability)]
protected theorem const_smul {𝕜} [SMul 𝕜 β] [ContinuousConstSMul 𝕜 β]
    (hf : AEStronglyMeasurable f μ) (c : 𝕜) : AEStronglyMeasurable (c • f) μ 
goal: MeasureTheory.AEStronglyMeasurable (c +ᵥ f) μ

full_name: MeasureTheory.AEStronglyMeasurable.const_smul
code: @[to_additive (attr := measurability)]
protected theorem const_smul {𝕜} [SMul 𝕜 β] [ContinuousConstSMul 𝕜 β]
    (hf : AEStronglyMeasurable f μ) (c : 𝕜) : AEStronglyMeasurable (c • f) μ 
goal: MeasureTheory.AEStronglyMeasurable (c • f) μ

full_name: MeasureTheory.AEStronglyMeasurable.const_vadd'
code: @[to_additive (attr := measurability)]
protected theorem const_smul' {𝕜} [SMul 𝕜 β] [ContinuousConstSMul 𝕜 β]
    (hf : AEStronglyMeasurable f μ) (c : 𝕜) : AEStronglyMeasurable (fun x => c • f x) μ 
goal: MeasureTheory.AEStronglyMeasurable (fun x => c +ᵥ f x) μ

full_name: MeasureTheory.AEStronglyMeasurable.const_smul'
code: @[to_additive (attr := measurability)]
protected theorem const_smul' {𝕜} [SMul 𝕜 β] [ContinuousConstSMul 𝕜 β]
    (hf : AEStronglyMeasurable f μ) (c : 𝕜) : AEStronglyMeasurable (fun x => c • f x) μ 
goal: MeasureTheory.AEStronglyMeasurable (fun x => c • f x) μ

full_name: MeasureTheory.AEStronglyMeasurable.vadd_const
code: @[to_additive (attr := measurability)]
protected theorem smul_const {𝕜} [TopologicalSpace 𝕜] [SMul 𝕜 β] [ContinuousSMul 𝕜 β] {f : α → 𝕜}
    (hf : AEStronglyMeasurable f μ) (c : β) : AEStronglyMeasurable (fun x => f x • c) μ 
goal: MeasureTheory.AEStronglyMeasurable (fun x => f x +ᵥ c) μ

full_name: MeasureTheory.AEStronglyMeasurable.smul_const
code: @[to_additive (attr := measurability)]
protected theorem smul_const {𝕜} [TopologicalSpace 𝕜] [SMul 𝕜 β] [ContinuousSMul 𝕜 β] {f : α → 𝕜}
    (hf : AEStronglyMeasurable f μ) (c : β) : AEStronglyMeasurable (fun x => f x • c) μ 
goal: MeasureTheory.AEStronglyMeasurable (fun x => f x • c) μ

full_name: MeasureTheory.AEStronglyMeasurable.sup
code: @[aesop safe 20 apply (rule_sets := [Measurable])]
protected theorem sup [SemilatticeSup β] [ContinuousSup β] (hf : AEStronglyMeasurable f μ)
    (hg : AEStronglyMeasurable g μ) : AEStronglyMeasurable (f ⊔ g) μ 
goal: MeasureTheory.AEStronglyMeasurable (f ⊔ g) μ

full_name: MeasureTheory.AEStronglyMeasurable.inf
code: @[aesop safe 20 apply (rule_sets := [Measurable])]
protected theorem inf [SemilatticeInf β] [ContinuousInf β] (hf : AEStronglyMeasurable f μ)
    (hg : AEStronglyMeasurable g μ) : AEStronglyMeasurable (f ⊓ g) μ 
goal: MeasureTheory.AEStronglyMeasurable (f ⊓ g) μ

full_name: List.aestronglyMeasurable_sum'
code: @[to_additive (attr := measurability)]
theorem _root_.List.aestronglyMeasurable_prod' (l : List (α → M))
    (hl : ∀ f ∈ l, AEStronglyMeasurable f μ) : AEStronglyMeasurable l.prod μ 
goal: MeasureTheory.AEStronglyMeasurable l.sum μ

full_name: List.aestronglyMeasurable_prod'
code: @[to_additive (attr := measurability)]
theorem _root_.List.aestronglyMeasurable_prod' (l : List (α → M))
    (hl : ∀ f ∈ l, AEStronglyMeasurable f μ) : AEStronglyMeasurable l.prod μ 
goal: MeasureTheory.AEStronglyMeasurable l.prod μ

full_name: List.aestronglyMeasurable_sum
code: @[to_additive (attr := measurability)]
theorem _root_.List.aestronglyMeasurable_prod
    (l : List (α → M)) (hl : ∀ f ∈ l, AEStronglyMeasurable f μ) :
    AEStronglyMeasurable (fun x => (l.map fun f : α → M => f x).prod) μ 
goal: MeasureTheory.AEStronglyMeasurable (fun x => (List.map (fun f => f x) l).sum) μ

full_name: List.aestronglyMeasurable_prod
code: @[to_additive (attr := measurability)]
theorem _root_.List.aestronglyMeasurable_prod
    (l : List (α → M)) (hl : ∀ f ∈ l, AEStronglyMeasurable f μ) :
    AEStronglyMeasurable (fun x => (l.map fun f : α → M => f x).prod) μ 
goal: MeasureTheory.AEStronglyMeasurable (fun x => (List.map (fun f => f x) l).prod) μ

full_name: Multiset.aestronglyMeasurable_prod'
code: @[to_additive (attr := measurability)]
theorem _root_.Multiset.aestronglyMeasurable_prod' (l : Multiset (α → M))
    (hl : ∀ f ∈ l, AEStronglyMeasurable f μ) : AEStronglyMeasurable l.prod μ 
goal: MeasureTheory.AEStronglyMeasurable l.prod μ

full_name: Multiset.aestronglyMeasurable_sum'
code: @[to_additive (attr := measurability)]
theorem _root_.Multiset.aestronglyMeasurable_prod' (l : Multiset (α → M))
    (hl : ∀ f ∈ l, AEStronglyMeasurable f μ) : AEStronglyMeasurable l.prod μ 
goal: MeasureTheory.AEStronglyMeasurable l.sum μ

full_name: Multiset.aestronglyMeasurable_prod
code: @[to_additive (attr := measurability)]
theorem _root_.Multiset.aestronglyMeasurable_prod (s : Multiset (α → M))
    (hs : ∀ f ∈ s, AEStronglyMeasurable f μ) :
    AEStronglyMeasurable (fun x => (s.map fun f : α → M => f x).prod) μ 
goal: MeasureTheory.AEStronglyMeasurable (fun x => (Multiset.map (fun f => f x) s).prod) μ

full_name: Multiset.aestronglyMeasurable_sum
code: @[to_additive (attr := measurability)]
theorem _root_.Multiset.aestronglyMeasurable_prod (s : Multiset (α → M))
    (hs : ∀ f ∈ s, AEStronglyMeasurable f μ) :
    AEStronglyMeasurable (fun x => (s.map fun f : α → M => f x).prod) μ 
goal: MeasureTheory.AEStronglyMeasurable (fun x => (Multiset.map (fun f => f x) s).sum) μ

full_name: Finset.aestronglyMeasurable_sum'
code: @[to_additive (attr := measurability)]
theorem _root_.Finset.aestronglyMeasurable_prod' {ι : Type*} {f : ι → α → M} (s : Finset ι)
    (hf : ∀ i ∈ s, AEStronglyMeasurable (f i) μ) : AEStronglyMeasurable (∏ i ∈ s, f i) μ 
goal: MeasureTheory.AEStronglyMeasurable (∑ i ∈ s, f i) μ

full_name: Finset.aestronglyMeasurable_prod'
code: @[to_additive (attr := measurability)]
theorem _root_.Finset.aestronglyMeasurable_prod' {ι : Type*} {f : ι → α → M} (s : Finset ι)
    (hf : ∀ i ∈ s, AEStronglyMeasurable (f i) μ) : AEStronglyMeasurable (∏ i ∈ s, f i) μ 
goal: MeasureTheory.AEStronglyMeasurable (∏ i ∈ s, f i) μ

full_name: Finset.aestronglyMeasurable_prod
code: @[to_additive (attr := measurability)]
theorem _root_.Finset.aestronglyMeasurable_prod {ι : Type*} {f : ι → α → M} (s : Finset ι)
    (hf : ∀ i ∈ s, AEStronglyMeasurable (f i) μ) :
    AEStronglyMeasurable (fun a => ∏ i ∈ s, f i a) μ 
goal: MeasureTheory.AEStronglyMeasurable (fun a => ∏ i ∈ s, f i a) μ

full_name: Finset.aestronglyMeasurable_sum
code: @[to_additive (attr := measurability)]
theorem _root_.Finset.aestronglyMeasurable_prod {ι : Type*} {f : ι → α → M} (s : Finset ι)
    (hf : ∀ i ∈ s, AEStronglyMeasurable (f i) μ) :
    AEStronglyMeasurable (fun a => ∏ i ∈ s, f i a) μ 
goal: MeasureTheory.AEStronglyMeasurable (fun a => ∑ i ∈ s, f i a) μ

full_name: AEMeasurable.aestronglyMeasurable
code: /-- In a space with second countable topology, measurable implies strongly measurable. -/
@[aesop 90% apply (rule_sets := [Measurable])]
theorem _root_.AEMeasurable.aestronglyMeasurable [PseudoMetrizableSpace β] [OpensMeasurableSpace β]
    [SecondCountableTopology β] (hf : AEMeasurable f μ) : AEStronglyMeasurable f μ 
goal: MeasureTheory.AEStronglyMeasurable f μ

full_name: aestronglyMeasurable_id
code: @[measurability]
theorem _root_.aestronglyMeasurable_id {α : Type*} [TopologicalSpace α] [PseudoMetrizableSpace α]
    {_ : MeasurableSpace α} [OpensMeasurableSpace α] [SecondCountableTopology α] {μ : Measure α} :
    AEStronglyMeasurable (id : α → α) μ 
goal: ∀ {x : MeasurableSpace α} [inst : OpensMeasurableSpace α] [inst : SecondCountableTopology α]
  {μ : MeasureTheory.Measure α}, MeasureTheory.AEStronglyMeasurable id μ

full_name: aestronglyMeasurable_iff_aemeasurable
code: /-- In a space with second countable topology, strongly measurable and measurable are equivalent. -/
theorem _root_.aestronglyMeasurable_iff_aemeasurable [PseudoMetrizableSpace β] [BorelSpace β]
    [SecondCountableTopology β] : AEStronglyMeasurable f μ ↔ AEMeasurable f μ 
goal: MeasureTheory.AEStronglyMeasurable f μ ↔ AEMeasurable f μ

full_name: MeasureTheory.AEStronglyMeasurable.dist
code: @[aesop safe 20 apply (rule_sets := [Measurable])]
protected theorem dist {β : Type*} [PseudoMetricSpace β] {f g : α → β}
    (hf : AEStronglyMeasurable f μ) (hg : AEStronglyMeasurable g μ) :
    AEStronglyMeasurable (fun x => dist (f x) (g x)) μ 
goal: MeasureTheory.AEStronglyMeasurable (fun x => dist (f x) (g x)) μ

full_name: MeasureTheory.AEStronglyMeasurable.norm
code: @[measurability]
protected theorem norm {β : Type*} [SeminormedAddCommGroup β] {f : α → β}
    (hf : AEStronglyMeasurable f μ) : AEStronglyMeasurable (fun x => ‖f x‖) μ 
goal: MeasureTheory.AEStronglyMeasurable (fun x => ‖f x‖) μ

full_name: MeasureTheory.AEStronglyMeasurable.nnnorm
code: @[measurability]
protected theorem nnnorm {β : Type*} [SeminormedAddCommGroup β] {f : α → β}
    (hf : AEStronglyMeasurable f μ) : AEStronglyMeasurable (fun x => ‖f x‖₊) μ 
goal: MeasureTheory.AEStronglyMeasurable (fun x => ‖f x‖₊) μ

full_name: MeasureTheory.AEStronglyMeasurable.ennnorm
code: @[measurability]
protected theorem ennnorm {β : Type*} [SeminormedAddCommGroup β] {f : α → β}
    (hf : AEStronglyMeasurable f μ) : AEMeasurable (fun a => (‖f a‖₊ : ℝ≥0∞)) μ 
goal: AEMeasurable (fun a => ↑‖f a‖₊) μ

full_name: MeasureTheory.AEStronglyMeasurable.edist
code: @[aesop safe 20 apply (rule_sets := [Measurable])]
protected theorem edist {β : Type*} [SeminormedAddCommGroup β] {f g : α → β}
    (hf : AEStronglyMeasurable f μ) (hg : AEStronglyMeasurable g μ) :
    AEMeasurable (fun a => edist (f a) (g a)) μ 
goal: AEMeasurable (fun a => edist (f a) (g a)) μ

full_name: MeasureTheory.AEStronglyMeasurable.real_toNNReal
code: @[measurability]
protected theorem real_toNNReal {f : α → ℝ} (hf : AEStronglyMeasurable f μ) :
    AEStronglyMeasurable (fun x => (f x).toNNReal) μ 
goal: MeasureTheory.AEStronglyMeasurable (fun x => (f x).toNNReal) μ

full_name: aestronglyMeasurable_indicator_iff
code: theorem _root_.aestronglyMeasurable_indicator_iff [Zero β] {s : Set α} (hs : MeasurableSet s) :
    AEStronglyMeasurable (indicator s f) μ ↔ AEStronglyMeasurable f (μ.restrict s) 
goal: MeasureTheory.AEStronglyMeasurable (s.indicator f) μ ↔ MeasureTheory.AEStronglyMeasurable f (μ.restrict s)

full_name: MeasureTheory.AEStronglyMeasurable.indicator
code: @[measurability]
protected theorem indicator [Zero β] (hfm : AEStronglyMeasurable f μ) {s : Set α}
    (hs : MeasurableSet s) : AEStronglyMeasurable (s.indicator f) μ 
goal: MeasureTheory.AEStronglyMeasurable (s.indicator f) μ

full_name: MeasureTheory.AEStronglyMeasurable.nullMeasurableSet_eq_fun
code: theorem nullMeasurableSet_eq_fun {E} [TopologicalSpace E] [MetrizableSpace E] {f g : α → E}
    (hf : AEStronglyMeasurable f μ) (hg : AEStronglyMeasurable g μ) :
    NullMeasurableSet { x | f x = g x } μ 
goal: MeasureTheory.NullMeasurableSet {x | f x = g x} μ

full_name: MeasureTheory.AEStronglyMeasurable.nullMeasurableSet_support
code: @[to_additive]
lemma nullMeasurableSet_mulSupport {E} [TopologicalSpace E] [MetrizableSpace E] [One E] {f : α → E}
    (hf : AEStronglyMeasurable f μ) : NullMeasurableSet (mulSupport f) μ 
goal: MeasureTheory.NullMeasurableSet (Function.support f) μ

full_name: MeasureTheory.AEStronglyMeasurable.nullMeasurableSet_mulSupport
code: @[to_additive]
lemma nullMeasurableSet_mulSupport {E} [TopologicalSpace E] [MetrizableSpace E] [One E] {f : α → E}
    (hf : AEStronglyMeasurable f μ) : NullMeasurableSet (mulSupport f) μ 
goal: MeasureTheory.NullMeasurableSet (Function.mulSupport f) μ

full_name: MeasureTheory.AEStronglyMeasurable.nullMeasurableSet_lt
code: theorem nullMeasurableSet_lt [LinearOrder β] [OrderClosedTopology β] [PseudoMetrizableSpace β]
    {f g : α → β} (hf : AEStronglyMeasurable f μ) (hg : AEStronglyMeasurable g μ) :
    NullMeasurableSet { a | f a < g a } μ 
goal: MeasureTheory.NullMeasurableSet {a | f a < g a} μ

full_name: MeasureTheory.AEStronglyMeasurable.nullMeasurableSet_le
code: theorem nullMeasurableSet_le [Preorder β] [OrderClosedTopology β] [PseudoMetrizableSpace β]
    {f g : α → β} (hf : AEStronglyMeasurable f μ) (hg : AEStronglyMeasurable g μ) :
    NullMeasurableSet { a | f a ≤ g a } μ 
goal: MeasureTheory.NullMeasurableSet {a | f a ≤ g a} μ

full_name: aestronglyMeasurable_of_aestronglyMeasurable_trim
code: theorem _root_.aestronglyMeasurable_of_aestronglyMeasurable_trim {α} {m m0 : MeasurableSpace α}
    {μ : Measure α} (hm : m ≤ m0) {f : α → β} (hf : AEStronglyMeasurable f (μ.trim hm)) :
    AEStronglyMeasurable f μ 
goal: MeasureTheory.AEStronglyMeasurable f μ

full_name: MeasureTheory.AEStronglyMeasurable.comp_aemeasurable
code: theorem comp_aemeasurable {γ : Type*} {_ : MeasurableSpace γ} {_ : MeasurableSpace α} {f : γ → α}
    {μ : Measure γ} (hg : AEStronglyMeasurable g (Measure.map f μ)) (hf : AEMeasurable f μ) :
    AEStronglyMeasurable (g ∘ f) μ 
goal: ∀ {x : MeasurableSpace γ} {x_1 : MeasurableSpace α} {f : γ → α} {μ : MeasureTheory.Measure γ},
  MeasureTheory.AEStronglyMeasurable g (MeasureTheory.Measure.map f μ) →
    AEMeasurable f μ → MeasureTheory.AEStronglyMeasurable (g ∘ f) μ

full_name: MeasureTheory.AEStronglyMeasurable.comp_measurable
code: theorem comp_measurable {γ : Type*} {_ : MeasurableSpace γ} {_ : MeasurableSpace α} {f : γ → α}
    {μ : Measure γ} (hg : AEStronglyMeasurable g (Measure.map f μ)) (hf : Measurable f) :
    AEStronglyMeasurable (g ∘ f) μ 
goal: ∀ {x : MeasurableSpace γ} {x_1 : MeasurableSpace α} {f : γ → α} {μ : MeasureTheory.Measure γ},
  MeasureTheory.AEStronglyMeasurable g (MeasureTheory.Measure.map f μ) →
    Measurable f → MeasureTheory.AEStronglyMeasurable (g ∘ f) μ

full_name: MeasureTheory.AEStronglyMeasurable.comp_quasiMeasurePreserving
code: theorem comp_quasiMeasurePreserving {γ : Type*} {_ : MeasurableSpace γ} {_ : MeasurableSpace α}
    {f : γ → α} {μ : Measure γ} {ν : Measure α} (hg : AEStronglyMeasurable g ν)
    (hf : QuasiMeasurePreserving f μ ν) : AEStronglyMeasurable (g ∘ f) μ 
goal: ∀ {x : MeasurableSpace γ} {x_1 : MeasurableSpace α} {f : γ → α} {μ : MeasureTheory.Measure γ}
  {ν : MeasureTheory.Measure α},
  MeasureTheory.AEStronglyMeasurable g ν →
    MeasureTheory.Measure.QuasiMeasurePreserving f μ ν → MeasureTheory.AEStronglyMeasurable (g ∘ f) μ

full_name: MeasureTheory.AEStronglyMeasurable.isSeparable_ae_range
code: theorem isSeparable_ae_range (hf : AEStronglyMeasurable f μ) :
    ∃ t : Set β, IsSeparable t ∧ ∀ᵐ x ∂μ, f x ∈ t 
goal: ∃ t, TopologicalSpace.IsSeparable t ∧ ∀ᵐ (x : α) ∂μ, f x ∈ t

full_name: aestronglyMeasurable_iff_aemeasurable_separable
code: /-- A function is almost everywhere strongly measurable if and only if it is almost everywhere
measurable, and up to a zero measure set its range is contained in a separable set. -/
theorem _root_.aestronglyMeasurable_iff_aemeasurable_separable [PseudoMetrizableSpace β]
    [MeasurableSpace β] [BorelSpace β] :
    AEStronglyMeasurable f μ ↔
      AEMeasurable f μ ∧ ∃ t : Set β, IsSeparable t ∧ ∀ᵐ x ∂μ, f x ∈ t 
goal: MeasureTheory.AEStronglyMeasurable f μ ↔ AEMeasurable f μ ∧ ∃ t, TopologicalSpace.IsSeparable t ∧ ∀ᵐ (x : α) ∂μ, f x ∈ t

full_name: aestronglyMeasurable_iff_nullMeasurable_separable
code: theorem _root_.aestronglyMeasurable_iff_nullMeasurable_separable [PseudoMetrizableSpace β]
    [MeasurableSpace β] [BorelSpace β] :
    AEStronglyMeasurable f μ ↔
      NullMeasurable f μ ∧ ∃ t : Set β, IsSeparable t ∧ ∀ᵐ x ∂μ, f x ∈ t 
goal: MeasureTheory.AEStronglyMeasurable f μ ↔
  MeasureTheory.NullMeasurable f μ ∧ ∃ t, TopologicalSpace.IsSeparable t ∧ ∀ᵐ (x : α) ∂μ, f x ∈ t

full_name: MeasurableEmbedding.aestronglyMeasurable_map_iff
code: theorem _root_.MeasurableEmbedding.aestronglyMeasurable_map_iff {γ : Type*}
    {mγ : MeasurableSpace γ} {mα : MeasurableSpace α} {f : γ → α} {μ : Measure γ}
    (hf : MeasurableEmbedding f) {g : α → β} :
    AEStronglyMeasurable g (Measure.map f μ) ↔ AEStronglyMeasurable (g ∘ f) μ 
goal: MeasureTheory.AEStronglyMeasurable g (MeasureTheory.Measure.map f μ) ↔ MeasureTheory.AEStronglyMeasurable (g ∘ f) μ

full_name: Embedding.aestronglyMeasurable_comp_iff
code: theorem _root_.Embedding.aestronglyMeasurable_comp_iff [PseudoMetrizableSpace β]
    [PseudoMetrizableSpace γ] {g : β → γ} {f : α → β} (hg : Embedding g) :
    AEStronglyMeasurable (fun x => g (f x)) μ ↔ AEStronglyMeasurable f μ 
goal: MeasureTheory.AEStronglyMeasurable (fun x => g (f x)) μ ↔ MeasureTheory.AEStronglyMeasurable f μ

full_name: aestronglyMeasurable_of_tendsto_ae
code: /-- An almost everywhere sequential limit of almost everywhere strongly measurable functions is
almost everywhere strongly measurable. -/
theorem _root_.aestronglyMeasurable_of_tendsto_ae {ι : Type*} [PseudoMetrizableSpace β]
    (u : Filter ι) [NeBot u] [IsCountablyGenerated u] {f : ι → α → β} {g : α → β}
    (hf : ∀ i, AEStronglyMeasurable (f i) μ) (lim : ∀ᵐ x ∂μ, Tendsto (fun n => f n x) u (𝓝 (g x))) :
    AEStronglyMeasurable g μ 
goal: MeasureTheory.AEStronglyMeasurable g μ

full_name: exists_stronglyMeasurable_limit_of_tendsto_ae
code: /-- If a sequence of almost everywhere strongly measurable functions converges almost everywhere,
one can select a strongly measurable function as the almost everywhere limit. -/
theorem _root_.exists_stronglyMeasurable_limit_of_tendsto_ae [PseudoMetrizableSpace β]
    {f : ℕ → α → β} (hf : ∀ n, AEStronglyMeasurable (f n) μ)
    (h_ae_tendsto : ∀ᵐ x ∂μ, ∃ l : β, Tendsto (fun n => f n x) atTop (𝓝 l)) :
    ∃ f_lim : α → β, StronglyMeasurable f_lim ∧
      ∀ᵐ x ∂μ, Tendsto (fun n => f n x) atTop (𝓝 (f_lim x)) 
goal: ∃ f_lim,
  MeasureTheory.StronglyMeasurable f_lim ∧ ∀ᵐ (x : α) ∂μ, Filter.Tendsto (fun n => f n x) Filter.atTop (𝓝 (f_lim x))

full_name: MeasureTheory.AEStronglyMeasurable.piecewise
code: theorem piecewise {s : Set α} [DecidablePred (· ∈ s)]
    (hs : MeasurableSet s) (hf : AEStronglyMeasurable f (μ.restrict s))
    (hg : AEStronglyMeasurable g (μ.restrict sᶜ)) :
    AEStronglyMeasurable (s.piecewise f g) μ 
goal: MeasureTheory.AEStronglyMeasurable (s.piecewise f g) μ

full_name: MeasureTheory.AEStronglyMeasurable.sum_measure
code: theorem sum_measure [PseudoMetrizableSpace β] {m : MeasurableSpace α} {μ : ι → Measure α}
    (h : ∀ i, AEStronglyMeasurable f (μ i)) : AEStronglyMeasurable f (Measure.sum μ) 
goal: MeasureTheory.AEStronglyMeasurable f (MeasureTheory.Measure.sum μ)

full_name: aestronglyMeasurable_sum_measure_iff
code: @[simp]
theorem _root_.aestronglyMeasurable_sum_measure_iff [PseudoMetrizableSpace β]
    {_m : MeasurableSpace α} {μ : ι → Measure α} :
    AEStronglyMeasurable f (sum μ) ↔ ∀ i, AEStronglyMeasurable f (μ i) 
goal: MeasureTheory.AEStronglyMeasurable f (MeasureTheory.Measure.sum μ) ↔
  ∀ (i : ι), MeasureTheory.AEStronglyMeasurable f (μ i)

full_name: aestronglyMeasurable_add_measure_iff
code: @[simp]
theorem _root_.aestronglyMeasurable_add_measure_iff [PseudoMetrizableSpace β] {ν : Measure α} :
    AEStronglyMeasurable f (μ + ν) ↔ AEStronglyMeasurable f μ ∧ AEStronglyMeasurable f ν 
goal: MeasureTheory.AEStronglyMeasurable f (μ + ν) ↔
  MeasureTheory.AEStronglyMeasurable f μ ∧ MeasureTheory.AEStronglyMeasurable f ν

full_name: MeasureTheory.AEStronglyMeasurable.add_measure
code: @[measurability]
theorem add_measure [PseudoMetrizableSpace β] {ν : Measure α} {f : α → β}
    (hμ : AEStronglyMeasurable f μ) (hν : AEStronglyMeasurable f ν) :
    AEStronglyMeasurable f (μ + ν) 
goal: MeasureTheory.AEStronglyMeasurable f (μ + ν)

full_name: MeasureTheory.AEStronglyMeasurable.iUnion
code: @[measurability]
protected theorem iUnion [PseudoMetrizableSpace β] {s : ι → Set α}
    (h : ∀ i, AEStronglyMeasurable f (μ.restrict (s i))) :
    AEStronglyMeasurable f (μ.restrict (⋃ i, s i)) 
goal: MeasureTheory.AEStronglyMeasurable f (μ.restrict (⋃ i, s i))

full_name: aestronglyMeasurable_iUnion_iff
code: @[simp]
theorem _root_.aestronglyMeasurable_iUnion_iff [PseudoMetrizableSpace β] {s : ι → Set α} :
    AEStronglyMeasurable f (μ.restrict (⋃ i, s i)) ↔
      ∀ i, AEStronglyMeasurable f (μ.restrict (s i)) 
goal: MeasureTheory.AEStronglyMeasurable f (μ.restrict (⋃ i, s i)) ↔
  ∀ (i : ι), MeasureTheory.AEStronglyMeasurable f (μ.restrict (s i))

full_name: aestronglyMeasurable_union_iff
code: @[simp]
theorem _root_.aestronglyMeasurable_union_iff [PseudoMetrizableSpace β] {s t : Set α} :
    AEStronglyMeasurable f (μ.restrict (s ∪ t)) ↔
      AEStronglyMeasurable f (μ.restrict s) ∧ AEStronglyMeasurable f (μ.restrict t) 
goal: MeasureTheory.AEStronglyMeasurable f (μ.restrict (s ∪ t)) ↔
  MeasureTheory.AEStronglyMeasurable f (μ.restrict s) ∧ MeasureTheory.AEStronglyMeasurable f (μ.restrict t)

full_name: MeasureTheory.AEStronglyMeasurable.aestronglyMeasurable_uIoc_iff
code: theorem aestronglyMeasurable_uIoc_iff [LinearOrder α] [PseudoMetrizableSpace β] {f : α → β}
    {a b : α} :
    AEStronglyMeasurable f (μ.restrict <| uIoc a b) ↔
      AEStronglyMeasurable f (μ.restrict <| Ioc a b) ∧
        AEStronglyMeasurable f (μ.restrict <| Ioc b a) 
goal: MeasureTheory.AEStronglyMeasurable f (μ.restrict (Ι a b)) ↔
  MeasureTheory.AEStronglyMeasurable f (μ.restrict (Set.Ioc a b)) ∧
    MeasureTheory.AEStronglyMeasurable f (μ.restrict (Set.Ioc b a))

full_name: MeasureTheory.AEStronglyMeasurable.smul_measure
code: @[measurability]
theorem smul_measure {R : Type*} [Monoid R] [DistribMulAction R ℝ≥0∞] [IsScalarTower R ℝ≥0∞ ℝ≥0∞]
    (h : AEStronglyMeasurable f μ) (c : R) : AEStronglyMeasurable f (c • μ) 
goal: MeasureTheory.AEStronglyMeasurable f (c • μ)

full_name: aestronglyMeasurable_const_smul_iff
code: theorem _root_.aestronglyMeasurable_const_smul_iff (c : G) :
    AEStronglyMeasurable (fun x => c • f x) μ ↔ AEStronglyMeasurable f μ 
goal: MeasureTheory.AEStronglyMeasurable (fun x => c • f x) μ ↔ MeasureTheory.AEStronglyMeasurable f μ

full_name: IsUnit.aestronglyMeasurable_const_smul_iff
code: nonrec theorem _root_.IsUnit.aestronglyMeasurable_const_smul_iff {c : M} (hc : IsUnit c) :
    AEStronglyMeasurable (fun x => c • f x) μ ↔ AEStronglyMeasurable f μ 
goal: MeasureTheory.AEStronglyMeasurable (fun x => c • f x) μ ↔ MeasureTheory.AEStronglyMeasurable f μ

full_name: aestronglyMeasurable_const_smul_iff₀
code: theorem _root_.aestronglyMeasurable_const_smul_iff₀ {c : G₀} (hc : c ≠ 0) :
    AEStronglyMeasurable (fun x => c • f x) μ ↔ AEStronglyMeasurable f μ 
goal: MeasureTheory.AEStronglyMeasurable (fun x => c • f x) μ ↔ MeasureTheory.AEStronglyMeasurable f μ

full_name: MeasureTheory.AEFinStronglyMeasurable.mk
code: /-- A `fin_strongly_measurable` function such that `f =ᵐ[μ] hf.mk f`. See lemmas
`fin_strongly_measurable_mk` and `ae_eq_mk`. -/
protected noncomputable def mk (f : α → β) (hf : AEFinStronglyMeasurable f μ) : α → β 
goal: α → β

full_name: MeasureTheory.AEFinStronglyMeasurable.finStronglyMeasurable_mk
code: theorem finStronglyMeasurable_mk (hf : AEFinStronglyMeasurable f μ) :
    FinStronglyMeasurable (hf.mk f) μ 
goal: MeasureTheory.FinStronglyMeasurable (MeasureTheory.AEFinStronglyMeasurable.mk f hf) μ

full_name: MeasureTheory.AEFinStronglyMeasurable.ae_eq_mk
code: theorem ae_eq_mk (hf : AEFinStronglyMeasurable f μ) : f =ᵐ[μ] hf.mk f 
goal: f =ᵐ[μ] MeasureTheory.AEFinStronglyMeasurable.mk f hf

full_name: MeasureTheory.AEFinStronglyMeasurable.aemeasurable
code: @[aesop 10% apply (rule_sets := [Measurable])]
protected theorem aemeasurable {β} [Zero β] [MeasurableSpace β] [TopologicalSpace β]
    [PseudoMetrizableSpace β] [BorelSpace β] {f : α → β} (hf : AEFinStronglyMeasurable f μ) :
    AEMeasurable f μ 
goal: AEMeasurable f μ

full_name: MeasureTheory.AEFinStronglyMeasurable.mul
code: @[aesop safe 20 (rule_sets := [Measurable])]
protected theorem mul [MonoidWithZero β] [ContinuousMul β] (hf : AEFinStronglyMeasurable f μ)
    (hg : AEFinStronglyMeasurable g μ) : AEFinStronglyMeasurable (f * g) μ 
goal: MeasureTheory.AEFinStronglyMeasurable (f * g) μ

full_name: MeasureTheory.AEFinStronglyMeasurable.add
code: @[aesop safe 20 (rule_sets := [Measurable])]
protected theorem add [AddMonoid β] [ContinuousAdd β] (hf : AEFinStronglyMeasurable f μ)
    (hg : AEFinStronglyMeasurable g μ) : AEFinStronglyMeasurable (f + g) μ 
goal: MeasureTheory.AEFinStronglyMeasurable (f + g) μ

full_name: MeasureTheory.AEFinStronglyMeasurable.neg
code: @[measurability]
protected theorem neg [AddGroup β] [TopologicalAddGroup β] (hf : AEFinStronglyMeasurable f μ) :
    AEFinStronglyMeasurable (-f) μ 
goal: MeasureTheory.AEFinStronglyMeasurable (-f) μ

full_name: MeasureTheory.AEFinStronglyMeasurable.sub
code: @[measurability]
protected theorem sub [AddGroup β] [ContinuousSub β] (hf : AEFinStronglyMeasurable f μ)
    (hg : AEFinStronglyMeasurable g μ) : AEFinStronglyMeasurable (f - g) μ 
goal: MeasureTheory.AEFinStronglyMeasurable (f - g) μ

full_name: MeasureTheory.AEFinStronglyMeasurable.const_smul
code: @[measurability]
protected theorem const_smul {𝕜} [TopologicalSpace 𝕜] [AddMonoid β] [Monoid 𝕜]
    [DistribMulAction 𝕜 β] [ContinuousSMul 𝕜 β] (hf : AEFinStronglyMeasurable f μ) (c : 𝕜) :
    AEFinStronglyMeasurable (c • f) μ 
goal: MeasureTheory.AEFinStronglyMeasurable (c • f) μ

full_name: MeasureTheory.AEFinStronglyMeasurable.sup
code: @[aesop safe 20 (rule_sets := [Measurable])]
protected theorem sup [SemilatticeSup β] [ContinuousSup β] (hf : AEFinStronglyMeasurable f μ)
    (hg : AEFinStronglyMeasurable g μ) : AEFinStronglyMeasurable (f ⊔ g) μ 
goal: MeasureTheory.AEFinStronglyMeasurable (f ⊔ g) μ

full_name: MeasureTheory.AEFinStronglyMeasurable.inf
code: @[aesop safe 20 (rule_sets := [Measurable])]
protected theorem inf [SemilatticeInf β] [ContinuousInf β] (hf : AEFinStronglyMeasurable f μ)
    (hg : AEFinStronglyMeasurable g μ) : AEFinStronglyMeasurable (f ⊓ g) μ 
goal: MeasureTheory.AEFinStronglyMeasurable (f ⊓ g) μ

full_name: MeasureTheory.AEFinStronglyMeasurable.exists_set_sigmaFinite
code: theorem exists_set_sigmaFinite (hf : AEFinStronglyMeasurable f μ) :
    ∃ t, MeasurableSet t ∧ f =ᵐ[μ.restrict tᶜ] 0 ∧ SigmaFinite (μ.restrict t) 
goal: ∃ t, MeasurableSet t ∧ f =ᵐ[μ.restrict tᶜ] 0 ∧ MeasureTheory.SigmaFinite (μ.restrict t)

full_name: MeasureTheory.AEFinStronglyMeasurable.sigmaFiniteSet
code: /-- A measurable set `t` such that `f =ᵐ[μ.restrict tᶜ] 0` and `sigma_finite (μ.restrict t)`. -/
def sigmaFiniteSet (hf : AEFinStronglyMeasurable f μ) : Set α 
goal: Set α

full_name: MeasureTheory.AEFinStronglyMeasurable.measurableSet
code: protected theorem measurableSet (hf : AEFinStronglyMeasurable f μ) :
    MeasurableSet hf.sigmaFiniteSet 
goal: MeasurableSet hf.sigmaFiniteSet

full_name: MeasureTheory.AEFinStronglyMeasurable.ae_eq_zero_compl
code: theorem ae_eq_zero_compl (hf : AEFinStronglyMeasurable f μ) :
    f =ᵐ[μ.restrict hf.sigmaFiniteSetᶜ] 0 
goal: f =ᵐ[μ.restrict hf.sigmaFiniteSetᶜ] 0

full_name: MeasureTheory.AEFinStronglyMeasurable.sigmaFinite_restrict
code: instance sigmaFinite_restrict (hf : AEFinStronglyMeasurable f μ) :
    SigmaFinite (μ.restrict hf.sigmaFiniteSet) 
goal: MeasureTheory.SigmaFinite (μ.restrict hf.sigmaFiniteSet)

full_name: MeasureTheory.finStronglyMeasurable_iff_measurable
code: /-- In a space with second countable topology and a sigma-finite measure, `FinStronglyMeasurable`
  and `Measurable` are equivalent. -/
theorem finStronglyMeasurable_iff_measurable {_m0 : MeasurableSpace α} (μ : Measure α)
    [SigmaFinite μ] : FinStronglyMeasurable f μ ↔ Measurable f 
goal: MeasureTheory.FinStronglyMeasurable f μ ↔ Measurable f

full_name: MeasureTheory.finStronglyMeasurable_of_measurable
code: /-- In a space with second countable topology and a sigma-finite measure, a measurable function
is `FinStronglyMeasurable`. -/
@[aesop 90% apply (rule_sets := [Measurable])]
theorem finStronglyMeasurable_of_measurable {_m0 : MeasurableSpace α} (μ : Measure α)
    [SigmaFinite μ] (hf : Measurable f) : FinStronglyMeasurable f μ 
goal: MeasureTheory.FinStronglyMeasurable f μ

full_name: MeasureTheory.aefinStronglyMeasurable_iff_aemeasurable
code: /-- In a space with second countable topology and a sigma-finite measure,
  `AEFinStronglyMeasurable` and `AEMeasurable` are equivalent. -/
theorem aefinStronglyMeasurable_iff_aemeasurable {_m0 : MeasurableSpace α} (μ : Measure α)
    [SigmaFinite μ] : AEFinStronglyMeasurable f μ ↔ AEMeasurable f μ 
goal: MeasureTheory.AEFinStronglyMeasurable f μ ↔ AEMeasurable f μ

full_name: MeasureTheory.aefinStronglyMeasurable_of_aemeasurable
code: /-- In a space with second countable topology and a sigma-finite measure,
  an `AEMeasurable` function is `AEFinStronglyMeasurable`. -/
@[aesop 90% apply (rule_sets := [Measurable])]
theorem aefinStronglyMeasurable_of_aemeasurable {_m0 : MeasurableSpace α} (μ : Measure α)
    [SigmaFinite μ] (hf : AEMeasurable f μ) : AEFinStronglyMeasurable f μ 
goal: MeasureTheory.AEFinStronglyMeasurable f μ

full_name: MeasureTheory.measurable_uncurry_of_continuous_of_measurable
code: theorem measurable_uncurry_of_continuous_of_measurable {α β ι : Type*} [TopologicalSpace ι]
    [MetrizableSpace ι] [MeasurableSpace ι] [SecondCountableTopology ι] [OpensMeasurableSpace ι]
    {mβ : MeasurableSpace β} [TopologicalSpace β] [PseudoMetrizableSpace β] [BorelSpace β]
    {m : MeasurableSpace α} {u : ι → α → β} (hu_cont : ∀ x, Continuous fun i => u i x)
    (h : ∀ i, Measurable (u i)) : Measurable (Function.uncurry u) 
goal: Measurable (Function.uncurry u)

full_name: MeasureTheory.stronglyMeasurable_uncurry_of_continuous_of_stronglyMeasurable
code: theorem stronglyMeasurable_uncurry_of_continuous_of_stronglyMeasurable {α β ι : Type*}
    [TopologicalSpace ι] [MetrizableSpace ι] [MeasurableSpace ι] [SecondCountableTopology ι]
    [OpensMeasurableSpace ι] [TopologicalSpace β] [PseudoMetrizableSpace β] [MeasurableSpace α]
    {u : ι → α → β} (hu_cont : ∀ x, Continuous fun i => u i x) (h : ∀ i, StronglyMeasurable (u i)) :
    StronglyMeasurable (Function.uncurry u) 
goal: MeasureTheory.StronglyMeasurable (Function.uncurry u)