File: Mathlib/MeasureTheory/Integral/Lebesgue.lean

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

full_name: MeasureTheory._aux_Mathlib_MeasureTheory_Integral_Lebesgue___unexpand_MeasureTheory_SimpleFunc_1
code: local infixr:25 " →ₛ " => SimpleFunc
goal: Lean.PrettyPrinter.Unexpander

full_name: MeasureTheory.wrapped._@.Mathlib.MeasureTheory.Integral.Lebesgue._hyg.464
code: /-- The **lower Lebesgue integral** of a function `f` with respect to a measure `μ`. -/
irreducible_def lintegral {_ : MeasurableSpace α} (μ : Measure α) (f : α → ℝ≥0∞) : ℝ≥0∞ 
goal: Subtype (Eq @MeasureTheory.definition✝)

full_name: MeasureTheory.lintegral_def
code: /-- The **lower Lebesgue integral** of a function `f` with respect to a measure `μ`. -/
irreducible_def lintegral {_ : MeasurableSpace α} (μ : Measure α) (f : α → ℝ≥0∞) : ℝ≥0∞ 
goal: ∀ {x : MeasurableSpace α} (μ : MeasureTheory.Measure α) (f : α → ℝ≥0∞),
  MeasureTheory.lintegral μ f = ⨆ g, ⨆ (_ : ↑g ≤ f), g.lintegral μ

full_name: MeasureTheory.lintegral
code: /-- The **lower Lebesgue integral** of a function `f` with respect to a measure `μ`. -/
irreducible_def lintegral {_ : MeasurableSpace α} (μ : Measure α) (f : α → ℝ≥0∞) : ℝ≥0∞ 
goal: {x : MeasurableSpace α} → MeasureTheory.Measure α → (α → ℝ≥0∞) → ℝ≥0∞

full_name: MeasureTheory.definition._@.Mathlib.MeasureTheory.Integral.Lebesgue._hyg.464
code: /-- The **lower Lebesgue integral** of a function `f` with respect to a measure `μ`. -/
irreducible_def lintegral {_ : MeasurableSpace α} (μ : Measure α) (f : α → ℝ≥0∞) : ℝ≥0∞ 
goal: {x : MeasurableSpace α} → MeasureTheory.Measure α → (α → ℝ≥0∞) → ℝ≥0∞

full_name: MeasureTheory.«_aux_Mathlib_MeasureTheory_Integral_Lebesgue___macroRules_MeasureTheory_term∫⁻_,_∂__1»
code: @[inherit_doc MeasureTheory.lintegral]
notation3 "∫⁻ "(...)", "r:60:(scoped f => f)" ∂"μ:70 => lintegral μ r
goal: Lean.Macro

full_name: MeasureTheory.«_aux_Mathlib_MeasureTheory_Integral_Lebesgue___macroRules_MeasureTheory_term∫⁻_,__1»
code: @[inherit_doc MeasureTheory.lintegral]
notation3 "∫⁻ "(...)", "r:60:(scoped f => lintegral volume f) => r
goal: Lean.Macro

full_name: MeasureTheory.«_aux_Mathlib_MeasureTheory_Integral_Lebesgue___macroRules_MeasureTheory_term∫⁻_In_,_∂__1»
code: @[inherit_doc MeasureTheory.lintegral]
notation3"∫⁻ "(...)" in "s", "r:60:(scoped f => f)" ∂"μ:70 => lintegral (Measure.restrict μ s) r
goal: Lean.Macro

full_name: MeasureTheory.«_aux_Mathlib_MeasureTheory_Integral_Lebesgue___macroRules_MeasureTheory_term∫⁻_In_,__1»
code: @[inherit_doc MeasureTheory.lintegral]
notation3"∫⁻ "(...)" in "s", "r:60:(scoped f => lintegral (Measure.restrict volume s) f) => r
goal: Lean.Macro

full_name: MeasureTheory.SimpleFunc.lintegral_eq_lintegral
code: theorem SimpleFunc.lintegral_eq_lintegral {m : MeasurableSpace α} (f : α →ₛ ℝ≥0∞) (μ : Measure α) :
    ∫⁻ a, f a ∂μ = f.lintegral μ 
goal: ∫⁻ (a : α), ↑f a ∂μ = f.lintegral μ

full_name: MeasureTheory.lintegral_mono'
code: @[mono]
theorem lintegral_mono' {m : MeasurableSpace α} ⦃μ ν : Measure α⦄ (hμν : μ ≤ ν) ⦃f g : α → ℝ≥0∞⦄
    (hfg : f ≤ g) : ∫⁻ a, f a ∂μ ≤ ∫⁻ a, g a ∂ν 
goal: ∫⁻ (a : α), f a ∂μ ≤ ∫⁻ (a : α), g a ∂ν

full_name: MeasureTheory.lintegral_mono_fn'
code: @[gcongr] theorem lintegral_mono_fn' ⦃f g : α → ℝ≥0∞⦄ (hfg : ∀ x, f x ≤ g x) (h2 : μ ≤ ν) :
    lintegral μ f ≤ lintegral ν g 
goal: MeasureTheory.lintegral μ f ≤ MeasureTheory.lintegral ν g

full_name: MeasureTheory.lintegral_mono
code: theorem lintegral_mono ⦃f g : α → ℝ≥0∞⦄ (hfg : f ≤ g) : ∫⁻ a, f a ∂μ ≤ ∫⁻ a, g a ∂μ 
goal: ∫⁻ (a : α), f a ∂μ ≤ ∫⁻ (a : α), g a ∂μ

full_name: MeasureTheory.lintegral_mono_fn
code: @[gcongr] theorem lintegral_mono_fn ⦃f g : α → ℝ≥0∞⦄ (hfg : ∀ x, f x ≤ g x) :
    lintegral μ f ≤ lintegral μ g 
goal: MeasureTheory.lintegral μ f ≤ MeasureTheory.lintegral μ g

full_name: MeasureTheory.lintegral_mono_nnreal
code: theorem lintegral_mono_nnreal {f g : α → ℝ≥0} (h : f ≤ g) : ∫⁻ a, f a ∂μ ≤ ∫⁻ a, g a ∂μ 
goal: ∫⁻ (a : α), ↑(f a) ∂μ ≤ ∫⁻ (a : α), ↑(g a) ∂μ

full_name: MeasureTheory.iSup_lintegral_measurable_le_eq_lintegral
code: theorem iSup_lintegral_measurable_le_eq_lintegral (f : α → ℝ≥0∞) :
    ⨆ (g : α → ℝ≥0∞) (_ : Measurable g) (_ : g ≤ f), ∫⁻ a, g a ∂μ = ∫⁻ a, f a ∂μ 
goal: ⨆ g, ⨆ (_ : Measurable g), ⨆ (_ : g ≤ f), ∫⁻ (a : α), g a ∂μ = ∫⁻ (a : α), f a ∂μ

full_name: MeasureTheory.lintegral_mono_set
code: theorem lintegral_mono_set {_ : MeasurableSpace α} ⦃μ : Measure α⦄ {s t : Set α} {f : α → ℝ≥0∞}
    (hst : s ⊆ t) : ∫⁻ x in s, f x ∂μ ≤ ∫⁻ x in t, f x ∂μ 
goal: ∀ {x : MeasurableSpace α} ⦃μ : MeasureTheory.Measure α⦄ {s t : Set α} {f : α → ℝ≥0∞},
  s ⊆ t → ∫⁻ (x : α) in s, f x ∂μ ≤ ∫⁻ (x : α) in t, f x ∂μ

full_name: MeasureTheory.lintegral_mono_set'
code: theorem lintegral_mono_set' {_ : MeasurableSpace α} ⦃μ : Measure α⦄ {s t : Set α} {f : α → ℝ≥0∞}
    (hst : s ≤ᵐ[μ] t) : ∫⁻ x in s, f x ∂μ ≤ ∫⁻ x in t, f x ∂μ 
goal: ∀ {x : MeasurableSpace α} ⦃μ : MeasureTheory.Measure α⦄ {s t : Set α} {f : α → ℝ≥0∞},
  s ≤ᵐ[μ] t → ∫⁻ (x : α) in s, f x ∂μ ≤ ∫⁻ (x : α) in t, f x ∂μ

full_name: MeasureTheory.monotone_lintegral
code: theorem monotone_lintegral {_ : MeasurableSpace α} (μ : Measure α) : Monotone (lintegral μ) 
goal: ∀ {x : MeasurableSpace α} (μ : MeasureTheory.Measure α), Monotone (MeasureTheory.lintegral μ)

full_name: MeasureTheory.lintegral_const
code: @[simp]
theorem lintegral_const (c : ℝ≥0∞) : ∫⁻ _, c ∂μ = c * μ univ 
goal: ∫⁻ (x : α), c ∂μ = c * μ Set.univ

full_name: MeasureTheory.lintegral_zero
code: theorem lintegral_zero : ∫⁻ _ : α, 0 ∂μ = 0 
goal: ∫⁻ (x : α), 0 ∂μ = 0

full_name: MeasureTheory.lintegral_zero_fun
code: theorem lintegral_zero_fun : lintegral μ (0 : α → ℝ≥0∞) = 0 
goal: MeasureTheory.lintegral μ 0 = 0

full_name: MeasureTheory.lintegral_one
code: theorem lintegral_one : ∫⁻ _, (1 : ℝ≥0∞) ∂μ = μ univ 
goal: ∫⁻ (x : α), 1 ∂μ = μ Set.univ

full_name: MeasureTheory.setLIntegral_const
code: theorem setLIntegral_const (s : Set α) (c : ℝ≥0∞) : ∫⁻ _ in s, c ∂μ = c * μ s 
goal: ∫⁻ (x : α) in s, c ∂μ = c * μ s

full_name: MeasureTheory.setLIntegral_one
code: theorem setLIntegral_one (s) : ∫⁻ _ in s, 1 ∂μ = μ s 
goal: ∫⁻ (x : α) in s, 1 ∂μ = μ s

full_name: MeasureTheory.setLIntegral_const_lt_top
code: theorem setLIntegral_const_lt_top [IsFiniteMeasure μ] (s : Set α) {c : ℝ≥0∞} (hc : c ≠ ∞) :
    ∫⁻ _ in s, c ∂μ < ∞ 
goal: ∫⁻ (x : α) in s, c ∂μ < ⊤

full_name: MeasureTheory.lintegral_const_lt_top
code: theorem lintegral_const_lt_top [IsFiniteMeasure μ] {c : ℝ≥0∞} (hc : c ≠ ∞) : ∫⁻ _, c ∂μ < ∞ 
goal: ∫⁻ (x : α), c ∂μ < ⊤

full_name: MeasureTheory.exists_measurable_le_lintegral_eq
code: /-- For any function `f : α → ℝ≥0∞`, there exists a measurable function `g ≤ f` with the same
integral. -/
theorem exists_measurable_le_lintegral_eq (f : α → ℝ≥0∞) :
    ∃ g : α → ℝ≥0∞, Measurable g ∧ g ≤ f ∧ ∫⁻ a, f a ∂μ = ∫⁻ a, g a ∂μ 
goal: ∃ g, Measurable g ∧ g ≤ f ∧ ∫⁻ (a : α), f a ∂μ = ∫⁻ (a : α), g a ∂μ

full_name: MeasureTheory.lintegral_eq_nnreal
code: /-- `∫⁻ a in s, f a ∂μ` is defined as the supremum of integrals of simple functions
`φ : α →ₛ ℝ≥0∞` such that `φ ≤ f`. This lemma says that it suffices to take
functions `φ : α →ₛ ℝ≥0`. -/
theorem lintegral_eq_nnreal {m : MeasurableSpace α} (f : α → ℝ≥0∞) (μ : Measure α) :
    ∫⁻ a, f a ∂μ =
      ⨆ (φ : α →ₛ ℝ≥0) (_ : ∀ x, ↑(φ x) ≤ f x), (φ.map ((↑) : ℝ≥0 → ℝ≥0∞)).lintegral μ 
goal: ∫⁻ (a : α), f a ∂μ =
  ⨆ φ, ⨆ (_ : ∀ (x : α), ↑(↑φ x) ≤ f x), (MeasureTheory.SimpleFunc.map ENNReal.ofNNReal φ).lintegral μ

full_name: MeasureTheory.exists_simpleFunc_forall_lintegral_sub_lt_of_pos
code: theorem exists_simpleFunc_forall_lintegral_sub_lt_of_pos {f : α → ℝ≥0∞} (h : ∫⁻ x, f x ∂μ ≠ ∞)
    {ε : ℝ≥0∞} (hε : ε ≠ 0) :
    ∃ φ : α →ₛ ℝ≥0,
      (∀ x, ↑(φ x) ≤ f x) ∧
        ∀ ψ : α →ₛ ℝ≥0, (∀ x, ↑(ψ x) ≤ f x) → (map (↑) (ψ - φ)).lintegral μ < ε 
goal: ∃ φ,
  (∀ (x : α), ↑(↑φ x) ≤ f x) ∧
    ∀ (ψ : MeasureTheory.SimpleFunc α ℝ≥0),
      (∀ (x : α), ↑(↑ψ x) ≤ f x) → (MeasureTheory.SimpleFunc.map ENNReal.ofNNReal (ψ - φ)).lintegral μ < ε

full_name: MeasureTheory.iSup_lintegral_le
code: theorem iSup_lintegral_le {ι : Sort*} (f : ι → α → ℝ≥0∞) :
    ⨆ i, ∫⁻ a, f i a ∂μ ≤ ∫⁻ a, ⨆ i, f i a ∂μ 
goal: ⨆ i, ∫⁻ (a : α), f i a ∂μ ≤ ∫⁻ (a : α), ⨆ i, f i a ∂μ

full_name: MeasureTheory.iSup₂_lintegral_le
code: theorem iSup₂_lintegral_le {ι : Sort*} {ι' : ι → Sort*} (f : ∀ i, ι' i → α → ℝ≥0∞) :
    ⨆ (i) (j), ∫⁻ a, f i j a ∂μ ≤ ∫⁻ a, ⨆ (i) (j), f i j a ∂μ 
goal: ⨆ i, ⨆ j, ∫⁻ (a : α), f i j a ∂μ ≤ ∫⁻ (a : α), ⨆ i, ⨆ j, f i j a ∂μ

full_name: MeasureTheory.le_iInf_lintegral
code: theorem le_iInf_lintegral {ι : Sort*} (f : ι → α → ℝ≥0∞) :
    ∫⁻ a, ⨅ i, f i a ∂μ ≤ ⨅ i, ∫⁻ a, f i a ∂μ 
goal: ∫⁻ (a : α), ⨅ i, f i a ∂μ ≤ ⨅ i, ∫⁻ (a : α), f i a ∂μ

full_name: MeasureTheory.le_iInf₂_lintegral
code: theorem le_iInf₂_lintegral {ι : Sort*} {ι' : ι → Sort*} (f : ∀ i, ι' i → α → ℝ≥0∞) :
    ∫⁻ a, ⨅ (i) (h : ι' i), f i h a ∂μ ≤ ⨅ (i) (h : ι' i), ∫⁻ a, f i h a ∂μ 
goal: ∫⁻ (a : α), ⨅ i, ⨅ h, f i h a ∂μ ≤ ⨅ i, ⨅ h, ∫⁻ (a : α), f i h a ∂μ

full_name: MeasureTheory.lintegral_mono_ae
code: theorem lintegral_mono_ae {f g : α → ℝ≥0∞} (h : ∀ᵐ a ∂μ, f a ≤ g a) :
    ∫⁻ a, f a ∂μ ≤ ∫⁻ a, g a ∂μ 
goal: ∫⁻ (a : α), f a ∂μ ≤ ∫⁻ (a : α), g a ∂μ

full_name: MeasureTheory.setLIntegral_mono_ae
code: /-- Lebesgue integral over a set is monotone in function.

This version assumes that the upper estimate is an a.e. measurable function
and the estimate holds a.e. on the set.
See also `setLIntegral_mono_ae'` for a version that assumes measurability of the set
but assumes no regularity of either function. -/
theorem setLIntegral_mono_ae {s : Set α} {f g : α → ℝ≥0∞} (hg : AEMeasurable g (μ.restrict s))
    (hfg : ∀ᵐ x ∂μ, x ∈ s → f x ≤ g x) : ∫⁻ x in s, f x ∂μ ≤ ∫⁻ x in s, g x ∂μ 
goal: ∫⁻ (x : α) in s, f x ∂μ ≤ ∫⁻ (x : α) in s, g x ∂μ

full_name: MeasureTheory.setLIntegral_mono
code: theorem setLIntegral_mono {s : Set α} {f g : α → ℝ≥0∞} (hg : Measurable g)
    (hfg : ∀ x ∈ s, f x ≤ g x) : ∫⁻ x in s, f x ∂μ ≤ ∫⁻ x in s, g x ∂μ 
goal: ∫⁻ (x : α) in s, f x ∂μ ≤ ∫⁻ (x : α) in s, g x ∂μ

full_name: MeasureTheory.setLIntegral_mono_ae'
code: theorem setLIntegral_mono_ae' {s : Set α} {f g : α → ℝ≥0∞} (hs : MeasurableSet s)
    (hfg : ∀ᵐ x ∂μ, x ∈ s → f x ≤ g x) : ∫⁻ x in s, f x ∂μ ≤ ∫⁻ x in s, g x ∂μ 
goal: ∫⁻ (x : α) in s, f x ∂μ ≤ ∫⁻ (x : α) in s, g x ∂μ

full_name: MeasureTheory.setLIntegral_mono'
code: theorem setLIntegral_mono' {s : Set α} {f g : α → ℝ≥0∞} (hs : MeasurableSet s)
    (hfg : ∀ x ∈ s, f x ≤ g x) : ∫⁻ x in s, f x ∂μ ≤ ∫⁻ x in s, g x ∂μ 
goal: ∫⁻ (x : α) in s, f x ∂μ ≤ ∫⁻ (x : α) in s, g x ∂μ

full_name: MeasureTheory.setLIntegral_le_lintegral
code: theorem setLIntegral_le_lintegral (s : Set α) (f : α → ℝ≥0∞) :
    ∫⁻ x in s, f x ∂μ ≤ ∫⁻ x, f x ∂μ 
goal: ∫⁻ (x : α) in s, f x ∂μ ≤ ∫⁻ (x : α), f x ∂μ

full_name: MeasureTheory.lintegral_congr_ae
code: theorem lintegral_congr_ae {f g : α → ℝ≥0∞} (h : f =ᵐ[μ] g) : ∫⁻ a, f a ∂μ = ∫⁻ a, g a ∂μ 
goal: ∫⁻ (a : α), f a ∂μ = ∫⁻ (a : α), g a ∂μ

full_name: MeasureTheory.lintegral_congr
code: theorem lintegral_congr {f g : α → ℝ≥0∞} (h : ∀ a, f a = g a) : ∫⁻ a, f a ∂μ = ∫⁻ a, g a ∂μ 
goal: ∫⁻ (a : α), f a ∂μ = ∫⁻ (a : α), g a ∂μ

full_name: MeasureTheory.setLIntegral_congr
code: theorem setLIntegral_congr {f : α → ℝ≥0∞} {s t : Set α} (h : s =ᵐ[μ] t) :
    ∫⁻ x in s, f x ∂μ = ∫⁻ x in t, f x ∂μ 
goal: ∫⁻ (x : α) in s, f x ∂μ = ∫⁻ (x : α) in t, f x ∂μ

full_name: MeasureTheory.setLIntegral_congr_fun
code: theorem setLIntegral_congr_fun {f g : α → ℝ≥0∞} {s : Set α} (hs : MeasurableSet s)
    (hfg : ∀ᵐ x ∂μ, x ∈ s → f x = g x) : ∫⁻ x in s, f x ∂μ = ∫⁻ x in s, g x ∂μ 
goal: ∫⁻ (x : α) in s, f x ∂μ = ∫⁻ (x : α) in s, g x ∂μ

full_name: MeasureTheory.lintegral_ofReal_le_lintegral_nnnorm
code: theorem lintegral_ofReal_le_lintegral_nnnorm (f : α → ℝ) :
    ∫⁻ x, ENNReal.ofReal (f x) ∂μ ≤ ∫⁻ x, ‖f x‖₊ ∂μ 
goal: ∫⁻ (x : α), ENNReal.ofReal (f x) ∂μ ≤ ∫⁻ (x : α), ↑‖f x‖₊ ∂μ

full_name: MeasureTheory.lintegral_nnnorm_eq_of_ae_nonneg
code: theorem lintegral_nnnorm_eq_of_ae_nonneg {f : α → ℝ} (h_nonneg : 0 ≤ᵐ[μ] f) :
    ∫⁻ x, ‖f x‖₊ ∂μ = ∫⁻ x, ENNReal.ofReal (f x) ∂μ 
goal: ∫⁻ (x : α), ↑‖f x‖₊ ∂μ = ∫⁻ (x : α), ENNReal.ofReal (f x) ∂μ

full_name: MeasureTheory.lintegral_nnnorm_eq_of_nonneg
code: theorem lintegral_nnnorm_eq_of_nonneg {f : α → ℝ} (h_nonneg : 0 ≤ f) :
    ∫⁻ x, ‖f x‖₊ ∂μ = ∫⁻ x, ENNReal.ofReal (f x) ∂μ 
goal: ∫⁻ (x : α), ↑‖f x‖₊ ∂μ = ∫⁻ (x : α), ENNReal.ofReal (f x) ∂μ

full_name: MeasureTheory.lintegral_iSup
code: /-- **Monotone convergence theorem** -- sometimes called **Beppo-Levi convergence**.
See `lintegral_iSup_directed` for a more general form. -/
theorem lintegral_iSup {f : ℕ → α → ℝ≥0∞} (hf : ∀ n, Measurable (f n)) (h_mono : Monotone f) :
    ∫⁻ a, ⨆ n, f n a ∂μ = ⨆ n, ∫⁻ a, f n a ∂μ 
goal: ∫⁻ (a : α), ⨆ n, f n a ∂μ = ⨆ n, ∫⁻ (a : α), f n a ∂μ

full_name: MeasureTheory.lintegral_iSup'
code: /-- Monotone convergence theorem -- sometimes called Beppo-Levi convergence. Version with
ae_measurable functions. -/
theorem lintegral_iSup' {f : ℕ → α → ℝ≥0∞} (hf : ∀ n, AEMeasurable (f n) μ)
    (h_mono : ∀ᵐ x ∂μ, Monotone fun n => f n x) : ∫⁻ a, ⨆ n, f n a ∂μ = ⨆ n, ∫⁻ a, f n a ∂μ 
goal: ∫⁻ (a : α), ⨆ n, f n a ∂μ = ⨆ n, ∫⁻ (a : α), f n a ∂μ

full_name: MeasureTheory.lintegral_tendsto_of_tendsto_of_monotone
code: /-- Monotone convergence theorem expressed with limits -/
theorem lintegral_tendsto_of_tendsto_of_monotone {f : ℕ → α → ℝ≥0∞} {F : α → ℝ≥0∞}
    (hf : ∀ n, AEMeasurable (f n) μ) (h_mono : ∀ᵐ x ∂μ, Monotone fun n => f n x)
    (h_tendsto : ∀ᵐ x ∂μ, Tendsto (fun n => f n x) atTop (𝓝 <| F x)) :
    Tendsto (fun n => ∫⁻ x, f n x ∂μ) atTop (𝓝 <| ∫⁻ x, F x ∂μ) 
goal: Filter.Tendsto (fun n => ∫⁻ (x : α), f n x ∂μ) Filter.atTop (𝓝 (∫⁻ (x : α), F x ∂μ))

full_name: MeasureTheory.lintegral_eq_iSup_eapprox_lintegral
code: theorem lintegral_eq_iSup_eapprox_lintegral {f : α → ℝ≥0∞} (hf : Measurable f) :
    ∫⁻ a, f a ∂μ = ⨆ n, (eapprox f n).lintegral μ 
goal: ∫⁻ (a : α), f a ∂μ = ⨆ n, (MeasureTheory.SimpleFunc.eapprox f n).lintegral μ

full_name: MeasureTheory.exists_pos_setLIntegral_lt_of_measure_lt
code: /-- If `f` has finite integral, then `∫⁻ x in s, f x ∂μ` is absolutely continuous in `s`: it tends
to zero as `μ s` tends to zero. This lemma states this fact in terms of `ε` and `δ`. -/
theorem exists_pos_setLIntegral_lt_of_measure_lt {f : α → ℝ≥0∞} (h : ∫⁻ x, f x ∂μ ≠ ∞) {ε : ℝ≥0∞}
    (hε : ε ≠ 0) : ∃ δ > 0, ∀ s, μ s < δ → ∫⁻ x in s, f x ∂μ < ε 
goal: ∃ δ > 0, ∀ (s : Set α), μ s < δ → ∫⁻ (x : α) in s, f x ∂μ < ε

full_name: MeasureTheory.tendsto_setLIntegral_zero
code: /-- If `f` has finite integral, then `∫⁻ x in s, f x ∂μ` is absolutely continuous in `s`: it tends
to zero as `μ s` tends to zero. -/
theorem tendsto_setLIntegral_zero {ι} {f : α → ℝ≥0∞} (h : ∫⁻ x, f x ∂μ ≠ ∞) {l : Filter ι}
    {s : ι → Set α} (hl : Tendsto (μ ∘ s) l (𝓝 0)) :
    Tendsto (fun i => ∫⁻ x in s i, f x ∂μ) l (𝓝 0) 
goal: Filter.Tendsto (fun i => ∫⁻ (x : α) in s i, f x ∂μ) l (𝓝 0)

full_name: MeasureTheory.le_lintegral_add
code: /-- The sum of the lower Lebesgue integrals of two functions is less than or equal to the integral
of their sum. The other inequality needs one of these functions to be (a.e.-)measurable. -/
theorem le_lintegral_add (f g : α → ℝ≥0∞) :
    ∫⁻ a, f a ∂μ + ∫⁻ a, g a ∂μ ≤ ∫⁻ a, f a + g a ∂μ 
goal: ∫⁻ (a : α), f a ∂μ + ∫⁻ (a : α), g a ∂μ ≤ ∫⁻ (a : α), f a + g a ∂μ

full_name: MeasureTheory.lintegral_add_aux
code: theorem lintegral_add_aux {f g : α → ℝ≥0∞} (hf : Measurable f) (hg : Measurable g) :
    ∫⁻ a, f a + g a ∂μ = ∫⁻ a, f a ∂μ + ∫⁻ a, g a ∂μ 
goal: ∫⁻ (a : α), f a + g a ∂μ = ∫⁻ (a : α), f a ∂μ + ∫⁻ (a : α), g a ∂μ

full_name: MeasureTheory.lintegral_add_left
code: /-- If `f g : α → ℝ≥0∞` are two functions and one of them is (a.e.) measurable, then the Lebesgue
integral of `f + g` equals the sum of integrals. This lemma assumes that `f` is integrable, see also
`MeasureTheory.lintegral_add_right` and primed versions of these lemmas. -/
@[simp]
theorem lintegral_add_left {f : α → ℝ≥0∞} (hf : Measurable f) (g : α → ℝ≥0∞) :
    ∫⁻ a, f a + g a ∂μ = ∫⁻ a, f a ∂μ + ∫⁻ a, g a ∂μ 
goal: ∫⁻ (a : α), f a + g a ∂μ = ∫⁻ (a : α), f a ∂μ + ∫⁻ (a : α), g a ∂μ

full_name: MeasureTheory.lintegral_add_left'
code: theorem lintegral_add_left' {f : α → ℝ≥0∞} (hf : AEMeasurable f μ) (g : α → ℝ≥0∞) :
    ∫⁻ a, f a + g a ∂μ = ∫⁻ a, f a ∂μ + ∫⁻ a, g a ∂μ 
goal: ∫⁻ (a : α), f a + g a ∂μ = ∫⁻ (a : α), f a ∂μ + ∫⁻ (a : α), g a ∂μ

full_name: MeasureTheory.lintegral_add_right'
code: theorem lintegral_add_right' (f : α → ℝ≥0∞) {g : α → ℝ≥0∞} (hg : AEMeasurable g μ) :
    ∫⁻ a, f a + g a ∂μ = ∫⁻ a, f a ∂μ + ∫⁻ a, g a ∂μ 
goal: ∫⁻ (a : α), f a + g a ∂μ = ∫⁻ (a : α), f a ∂μ + ∫⁻ (a : α), g a ∂μ

full_name: MeasureTheory.lintegral_add_right
code: /-- If `f g : α → ℝ≥0∞` are two functions and one of them is (a.e.) measurable, then the Lebesgue
integral of `f + g` equals the sum of integrals. This lemma assumes that `g` is integrable, see also
`MeasureTheory.lintegral_add_left` and primed versions of these lemmas. -/
@[simp]
theorem lintegral_add_right (f : α → ℝ≥0∞) {g : α → ℝ≥0∞} (hg : Measurable g) :
    ∫⁻ a, f a + g a ∂μ = ∫⁻ a, f a ∂μ + ∫⁻ a, g a ∂μ 
goal: ∫⁻ (a : α), f a + g a ∂μ = ∫⁻ (a : α), f a ∂μ + ∫⁻ (a : α), g a ∂μ

full_name: MeasureTheory.lintegral_smul_measure
code: @[simp]
theorem lintegral_smul_measure (c : ℝ≥0∞) (f : α → ℝ≥0∞) : ∫⁻ a, f a ∂c • μ = c * ∫⁻ a, f a ∂μ 
goal: ∫⁻ (a : α), f a ∂c • μ = c * ∫⁻ (a : α), f a ∂μ

full_name: MeasureTheory.setLIntegral_smul_measure
code: lemma setLIntegral_smul_measure (c : ℝ≥0∞) (f : α → ℝ≥0∞) (s : Set α) :
    ∫⁻ a in s, f a ∂(c • μ) = c * ∫⁻ a in s, f a ∂μ 
goal: ∫⁻ (a : α) in s, f a ∂c • μ = c * ∫⁻ (a : α) in s, f a ∂μ

full_name: MeasureTheory.lintegral_sum_measure
code: @[simp]
theorem lintegral_sum_measure {m : MeasurableSpace α} {ι} (f : α → ℝ≥0∞) (μ : ι → Measure α) :
    ∫⁻ a, f a ∂Measure.sum μ = ∑' i, ∫⁻ a, f a ∂μ i 
goal: ∫⁻ (a : α), f a ∂MeasureTheory.Measure.sum μ = ∑' (i : ι), ∫⁻ (a : α), f a ∂μ i

full_name: MeasureTheory.hasSum_lintegral_measure
code: theorem hasSum_lintegral_measure {ι} {_ : MeasurableSpace α} (f : α → ℝ≥0∞) (μ : ι → Measure α) :
    HasSum (fun i => ∫⁻ a, f a ∂μ i) (∫⁻ a, f a ∂Measure.sum μ) 
goal: ∀ {x : MeasurableSpace α} (f : α → ℝ≥0∞) (μ : ι → MeasureTheory.Measure α),
  HasSum (fun i => ∫⁻ (a : α), f a ∂μ i) (∫⁻ (a : α), f a ∂MeasureTheory.Measure.sum μ)

full_name: MeasureTheory.lintegral_add_measure
code: @[simp]
theorem lintegral_add_measure {m : MeasurableSpace α} (f : α → ℝ≥0∞) (μ ν : Measure α) :
    ∫⁻ a, f a ∂(μ + ν) = ∫⁻ a, f a ∂μ + ∫⁻ a, f a ∂ν 
goal: ∫⁻ (a : α), f a ∂(μ + ν) = ∫⁻ (a : α), f a ∂μ + ∫⁻ (a : α), f a ∂ν

full_name: MeasureTheory.lintegral_finset_sum_measure
code: @[simp]
theorem lintegral_finset_sum_measure {ι} {m : MeasurableSpace α} (s : Finset ι) (f : α → ℝ≥0∞)
    (μ : ι → Measure α) : ∫⁻ a, f a ∂(∑ i ∈ s, μ i) = ∑ i ∈ s, ∫⁻ a, f a ∂μ i 
goal: ∫⁻ (a : α), f a ∂∑ i ∈ s, μ i = ∑ i ∈ s, ∫⁻ (a : α), f a ∂μ i

full_name: MeasureTheory.lintegral_zero_measure
code: @[simp]
theorem lintegral_zero_measure {m : MeasurableSpace α} (f : α → ℝ≥0∞) :
    ∫⁻ a, f a ∂(0 : Measure α) = 0 
goal: ∫⁻ (a : α), f a ∂0 = 0

full_name: MeasureTheory.lintegral_of_isEmpty
code: @[simp]
theorem lintegral_of_isEmpty {α} [MeasurableSpace α] [IsEmpty α] (μ : Measure α) (f : α → ℝ≥0∞) :
    ∫⁻ x, f x ∂μ = 0 
goal: ∫⁻ (x : α), f x ∂μ = 0

full_name: MeasureTheory.setLIntegral_empty
code: theorem setLIntegral_empty (f : α → ℝ≥0∞) : ∫⁻ x in ∅, f x ∂μ = 0 
goal: ∫⁻ (x : α) in ∅, f x ∂μ = 0

full_name: MeasureTheory.setLIntegral_univ
code: theorem setLIntegral_univ (f : α → ℝ≥0∞) : ∫⁻ x in univ, f x ∂μ = ∫⁻ x, f x ∂μ 
goal: ∫⁻ (x : α) in Set.univ, f x ∂μ = ∫⁻ (x : α), f x ∂μ

full_name: MeasureTheory.setLIntegral_measure_zero
code: theorem setLIntegral_measure_zero (s : Set α) (f : α → ℝ≥0∞) (hs' : μ s = 0) :
    ∫⁻ x in s, f x ∂μ = 0 
goal: ∫⁻ (x : α) in s, f x ∂μ = 0

full_name: MeasureTheory.lintegral_finset_sum'
code: theorem lintegral_finset_sum' (s : Finset β) {f : β → α → ℝ≥0∞}
    (hf : ∀ b ∈ s, AEMeasurable (f b) μ) :
    ∫⁻ a, ∑ b ∈ s, f b a ∂μ = ∑ b ∈ s, ∫⁻ a, f b a ∂μ 
goal: ∫⁻ (a : α), ∑ b ∈ s, f b a ∂μ = ∑ b ∈ s, ∫⁻ (a : α), f b a ∂μ

full_name: MeasureTheory.lintegral_finset_sum
code: theorem lintegral_finset_sum (s : Finset β) {f : β → α → ℝ≥0∞} (hf : ∀ b ∈ s, Measurable (f b)) :
    ∫⁻ a, ∑ b ∈ s, f b a ∂μ = ∑ b ∈ s, ∫⁻ a, f b a ∂μ 
goal: ∫⁻ (a : α), ∑ b ∈ s, f b a ∂μ = ∑ b ∈ s, ∫⁻ (a : α), f b a ∂μ

full_name: MeasureTheory.lintegral_const_mul
code: @[simp]
theorem lintegral_const_mul (r : ℝ≥0∞) {f : α → ℝ≥0∞} (hf : Measurable f) :
    ∫⁻ a, r * f a ∂μ = r * ∫⁻ a, f a ∂μ 
goal: ∫⁻ (a : α), r * f a ∂μ = r * ∫⁻ (a : α), f a ∂μ

full_name: MeasureTheory.lintegral_const_mul''
code: theorem lintegral_const_mul'' (r : ℝ≥0∞) {f : α → ℝ≥0∞} (hf : AEMeasurable f μ) :
    ∫⁻ a, r * f a ∂μ = r * ∫⁻ a, f a ∂μ 
goal: ∫⁻ (a : α), r * f a ∂μ = r * ∫⁻ (a : α), f a ∂μ

full_name: MeasureTheory.lintegral_const_mul_le
code: theorem lintegral_const_mul_le (r : ℝ≥0∞) (f : α → ℝ≥0∞) :
    r * ∫⁻ a, f a ∂μ ≤ ∫⁻ a, r * f a ∂μ 
goal: r * ∫⁻ (a : α), f a ∂μ ≤ ∫⁻ (a : α), r * f a ∂μ

full_name: MeasureTheory.lintegral_const_mul'
code: theorem lintegral_const_mul' (r : ℝ≥0∞) (f : α → ℝ≥0∞) (hr : r ≠ ∞) :
    ∫⁻ a, r * f a ∂μ = r * ∫⁻ a, f a ∂μ 
goal: ∫⁻ (a : α), r * f a ∂μ = r * ∫⁻ (a : α), f a ∂μ

full_name: MeasureTheory.lintegral_mul_const
code: theorem lintegral_mul_const (r : ℝ≥0∞) {f : α → ℝ≥0∞} (hf : Measurable f) :
    ∫⁻ a, f a * r ∂μ = (∫⁻ a, f a ∂μ) * r 
goal: ∫⁻ (a : α), f a * r ∂μ = (∫⁻ (a : α), f a ∂μ) * r

full_name: MeasureTheory.lintegral_mul_const''
code: theorem lintegral_mul_const'' (r : ℝ≥0∞) {f : α → ℝ≥0∞} (hf : AEMeasurable f μ) :
    ∫⁻ a, f a * r ∂μ = (∫⁻ a, f a ∂μ) * r 
goal: ∫⁻ (a : α), f a * r ∂μ = (∫⁻ (a : α), f a ∂μ) * r

full_name: MeasureTheory.lintegral_mul_const_le
code: theorem lintegral_mul_const_le (r : ℝ≥0∞) (f : α → ℝ≥0∞) :
    (∫⁻ a, f a ∂μ) * r ≤ ∫⁻ a, f a * r ∂μ 
goal: (∫⁻ (a : α), f a ∂μ) * r ≤ ∫⁻ (a : α), f a * r ∂μ

full_name: MeasureTheory.lintegral_mul_const'
code: theorem lintegral_mul_const' (r : ℝ≥0∞) (f : α → ℝ≥0∞) (hr : r ≠ ∞) :
    ∫⁻ a, f a * r ∂μ = (∫⁻ a, f a ∂μ) * r 
goal: ∫⁻ (a : α), f a * r ∂μ = (∫⁻ (a : α), f a ∂μ) * r

full_name: MeasureTheory.lintegral_lintegral_mul
code: theorem lintegral_lintegral_mul {β} [MeasurableSpace β] {ν : Measure β} {f : α → ℝ≥0∞}
    {g : β → ℝ≥0∞} (hf : AEMeasurable f μ) (hg : AEMeasurable g ν) :
    ∫⁻ x, ∫⁻ y, f x * g y ∂ν ∂μ = (∫⁻ x, f x ∂μ) * ∫⁻ y, g y ∂ν 
goal: ∫⁻ (x : α), ∫⁻ (y : β), f x * g y ∂ν ∂μ = (∫⁻ (x : α), f x ∂μ) * ∫⁻ (y : β), g y ∂ν

full_name: MeasureTheory.lintegral_rw₁
code: theorem lintegral_rw₁ {f f' : α → β} (h : f =ᵐ[μ] f') (g : β → ℝ≥0∞) :
    ∫⁻ a, g (f a) ∂μ = ∫⁻ a, g (f' a) ∂μ 
goal: ∫⁻ (a : α), g (f a) ∂μ = ∫⁻ (a : α), g (f' a) ∂μ

full_name: MeasureTheory.lintegral_rw₂
code: theorem lintegral_rw₂ {f₁ f₁' : α → β} {f₂ f₂' : α → γ} (h₁ : f₁ =ᵐ[μ] f₁') (h₂ : f₂ =ᵐ[μ] f₂')
    (g : β → γ → ℝ≥0∞) : ∫⁻ a, g (f₁ a) (f₂ a) ∂μ = ∫⁻ a, g (f₁' a) (f₂' a) ∂μ 
goal: ∫⁻ (a : α), g (f₁ a) (f₂ a) ∂μ = ∫⁻ (a : α), g (f₁' a) (f₂' a) ∂μ

full_name: MeasureTheory.lintegral_indicator_le
code: theorem lintegral_indicator_le (f : α → ℝ≥0∞) (s : Set α) :
    ∫⁻ a, s.indicator f a ∂μ ≤ ∫⁻ a in s, f a ∂μ 
goal: ∫⁻ (a : α), s.indicator f a ∂μ ≤ ∫⁻ (a : α) in s, f a ∂μ

full_name: MeasureTheory.lintegral_indicator
code: @[simp]
theorem lintegral_indicator (f : α → ℝ≥0∞) {s : Set α} (hs : MeasurableSet s) :
    ∫⁻ a, s.indicator f a ∂μ = ∫⁻ a in s, f a ∂μ 
goal: ∫⁻ (a : α), s.indicator f a ∂μ = ∫⁻ (a : α) in s, f a ∂μ

full_name: MeasureTheory.lintegral_indicator₀
code: theorem lintegral_indicator₀ (f : α → ℝ≥0∞) {s : Set α} (hs : NullMeasurableSet s μ) :
    ∫⁻ a, s.indicator f a ∂μ = ∫⁻ a in s, f a ∂μ 
goal: ∫⁻ (a : α), s.indicator f a ∂μ = ∫⁻ (a : α) in s, f a ∂μ

full_name: MeasureTheory.lintegral_indicator_const_le
code: theorem lintegral_indicator_const_le (s : Set α) (c : ℝ≥0∞) :
    ∫⁻ a, s.indicator (fun _ => c) a ∂μ ≤ c * μ s 
goal: ∫⁻ (a : α), s.indicator (fun x => c) a ∂μ ≤ c * μ s

full_name: MeasureTheory.lintegral_indicator_const₀
code: theorem lintegral_indicator_const₀ {s : Set α} (hs : NullMeasurableSet s μ) (c : ℝ≥0∞) :
    ∫⁻ a, s.indicator (fun _ => c) a ∂μ = c * μ s 
goal: ∫⁻ (a : α), s.indicator (fun x => c) a ∂μ = c * μ s

full_name: MeasureTheory.lintegral_indicator_const
code: theorem lintegral_indicator_const {s : Set α} (hs : MeasurableSet s) (c : ℝ≥0∞) :
    ∫⁻ a, s.indicator (fun _ => c) a ∂μ = c * μ s 
goal: ∫⁻ (a : α), s.indicator (fun x => c) a ∂μ = c * μ s

full_name: MeasureTheory.setLIntegral_eq_of_support_subset
code: lemma setLIntegral_eq_of_support_subset {s : Set α} {f : α → ℝ≥0∞} (hsf : f.support ⊆ s) :
    ∫⁻ x in s, f x ∂μ = ∫⁻ x, f x ∂μ 
goal: ∫⁻ (x : α) in s, f x ∂μ = ∫⁻ (x : α), f x ∂μ

full_name: MeasureTheory.setLIntegral_eq_const
code: theorem setLIntegral_eq_const {f : α → ℝ≥0∞} (hf : Measurable f) (r : ℝ≥0∞) :
    ∫⁻ x in { x | f x = r }, f x ∂μ = r * μ { x | f x = r } 
goal: ∫⁻ (x : α) in {x | f x = r}, f x ∂μ = r * μ {x | f x = r}

full_name: MeasureTheory.lintegral_indicator_one_le
code: theorem lintegral_indicator_one_le (s : Set α) : ∫⁻ a, s.indicator 1 a ∂μ ≤ μ s 
goal: ∫⁻ (a : α), s.indicator 1 a ∂μ ≤ μ s

full_name: MeasureTheory.lintegral_indicator_one₀
code: @[simp]
theorem lintegral_indicator_one₀ {s : Set α} (hs : NullMeasurableSet s μ) :
    ∫⁻ a, s.indicator 1 a ∂μ = μ s 
goal: ∫⁻ (a : α), s.indicator 1 a ∂μ = μ s

full_name: MeasureTheory.lintegral_indicator_one
code: @[simp]
theorem lintegral_indicator_one {s : Set α} (hs : MeasurableSet s) :
    ∫⁻ a, s.indicator 1 a ∂μ = μ s 
goal: ∫⁻ (a : α), s.indicator 1 a ∂μ = μ s

full_name: MeasureTheory.lintegral_add_mul_meas_add_le_le_lintegral
code: /-- A version of **Markov's inequality** for two functions. It doesn't follow from the standard
Markov's inequality because we only assume measurability of `g`, not `f`. -/
theorem lintegral_add_mul_meas_add_le_le_lintegral {f g : α → ℝ≥0∞} (hle : f ≤ᵐ[μ] g)
    (hg : AEMeasurable g μ) (ε : ℝ≥0∞) :
    ∫⁻ a, f a ∂μ + ε * μ { x | f x + ε ≤ g x } ≤ ∫⁻ a, g a ∂μ 
goal: ∫⁻ (a : α), f a ∂μ + ε * μ {x | f x + ε ≤ g x} ≤ ∫⁻ (a : α), g a ∂μ

full_name: MeasureTheory.mul_meas_ge_le_lintegral₀
code: /-- **Markov's inequality** also known as **Chebyshev's first inequality**. -/
theorem mul_meas_ge_le_lintegral₀ {f : α → ℝ≥0∞} (hf : AEMeasurable f μ) (ε : ℝ≥0∞) :
    ε * μ { x | ε ≤ f x } ≤ ∫⁻ a, f a ∂μ 
goal: ε * μ {x | ε ≤ f x} ≤ ∫⁻ (a : α), f a ∂μ

full_name: MeasureTheory.mul_meas_ge_le_lintegral
code: /-- **Markov's inequality** also known as **Chebyshev's first inequality**. For a version assuming
`AEMeasurable`, see `mul_meas_ge_le_lintegral₀`. -/
theorem mul_meas_ge_le_lintegral {f : α → ℝ≥0∞} (hf : Measurable f) (ε : ℝ≥0∞) :
    ε * μ { x | ε ≤ f x } ≤ ∫⁻ a, f a ∂μ 
goal: ε * μ {x | ε ≤ f x} ≤ ∫⁻ (a : α), f a ∂μ

full_name: MeasureTheory.meas_le_lintegral₀
code: lemma meas_le_lintegral₀ {f : α → ℝ≥0∞} (hf : AEMeasurable f μ)
    {s : Set α} (hs : ∀ x ∈ s, 1 ≤ f x) : μ s ≤ ∫⁻ a, f a ∂μ 
goal: μ s ≤ ∫⁻ (a : α), f a ∂μ

full_name: MeasureTheory.lintegral_le_meas
code: lemma lintegral_le_meas {s : Set α} {f : α → ℝ≥0∞} (hf : ∀ a, f a ≤ 1) (h'f : ∀ a ∈ sᶜ, f a = 0) :
    ∫⁻ a, f a ∂μ ≤ μ s 
goal: ∫⁻ (a : α), f a ∂μ ≤ μ s

full_name: MeasureTheory.lintegral_eq_top_of_measure_eq_top_ne_zero
code: theorem lintegral_eq_top_of_measure_eq_top_ne_zero {f : α → ℝ≥0∞} (hf : AEMeasurable f μ)
    (hμf : μ {x | f x = ∞} ≠ 0) : ∫⁻ x, f x ∂μ = ∞ 
goal: ∫⁻ (x : α), f x ∂μ = ⊤

full_name: MeasureTheory.setLintegral_eq_top_of_measure_eq_top_ne_zero
code: theorem setLintegral_eq_top_of_measure_eq_top_ne_zero {f : α → ℝ≥0∞} {s : Set α}
    (hf : AEMeasurable f (μ.restrict s)) (hμf : μ ({x ∈ s | f x = ∞}) ≠ 0) :
    ∫⁻ x in s, f x ∂μ = ∞ 
goal: ∫⁻ (x : α) in s, f x ∂μ = ⊤

full_name: MeasureTheory.measure_eq_top_of_lintegral_ne_top
code: theorem measure_eq_top_of_lintegral_ne_top {f : α → ℝ≥0∞}
    (hf : AEMeasurable f μ) (hμf : ∫⁻ x, f x ∂μ ≠ ∞) : μ {x | f x = ∞} = 0 
goal: μ {x | f x = ⊤} = 0

full_name: MeasureTheory.measure_eq_top_of_setLintegral_ne_top
code: theorem measure_eq_top_of_setLintegral_ne_top {f : α → ℝ≥0∞} {s : Set α}
    (hf : AEMeasurable f (μ.restrict s)) (hμf : ∫⁻ x in s, f x ∂μ ≠ ∞) :
    μ ({x ∈ s | f x = ∞}) = 0 
goal: μ {x | x ∈ s ∧ f x = ⊤} = 0

full_name: MeasureTheory.meas_ge_le_lintegral_div
code: /-- **Markov's inequality**, also known as **Chebyshev's first inequality**. -/
theorem meas_ge_le_lintegral_div {f : α → ℝ≥0∞} (hf : AEMeasurable f μ) {ε : ℝ≥0∞} (hε : ε ≠ 0)
    (hε' : ε ≠ ∞) : μ { x | ε ≤ f x } ≤ (∫⁻ a, f a ∂μ) / ε 
goal: μ {x | ε ≤ f x} ≤ (∫⁻ (a : α), f a ∂μ) / ε

full_name: MeasureTheory.ae_eq_of_ae_le_of_lintegral_le
code: theorem ae_eq_of_ae_le_of_lintegral_le {f g : α → ℝ≥0∞} (hfg : f ≤ᵐ[μ] g) (hf : ∫⁻ x, f x ∂μ ≠ ∞)
    (hg : AEMeasurable g μ) (hgf : ∫⁻ x, g x ∂μ ≤ ∫⁻ x, f x ∂μ) : f =ᵐ[μ] g 
goal: f =ᵐ[μ] g

full_name: MeasureTheory.lintegral_eq_zero_iff'
code: @[simp]
theorem lintegral_eq_zero_iff' {f : α → ℝ≥0∞} (hf : AEMeasurable f μ) :
    ∫⁻ a, f a ∂μ = 0 ↔ f =ᵐ[μ] 0 
goal: ∫⁻ (a : α), f a ∂μ = 0 ↔ f =ᵐ[μ] 0

full_name: MeasureTheory.lintegral_eq_zero_iff
code: @[simp]
theorem lintegral_eq_zero_iff {f : α → ℝ≥0∞} (hf : Measurable f) : ∫⁻ a, f a ∂μ = 0 ↔ f =ᵐ[μ] 0 
goal: ∫⁻ (a : α), f a ∂μ = 0 ↔ f =ᵐ[μ] 0

full_name: MeasureTheory.lintegral_pos_iff_support
code: theorem lintegral_pos_iff_support {f : α → ℝ≥0∞} (hf : Measurable f) :
    (0 < ∫⁻ a, f a ∂μ) ↔ 0 < μ (Function.support f) 
goal: 0 < ∫⁻ (a : α), f a ∂μ ↔ 0 < μ (Function.support f)

full_name: MeasureTheory.setLintegral_pos_iff
code: theorem setLintegral_pos_iff {f : α → ℝ≥0∞} (hf : Measurable f) {s : Set α} :
    0 < ∫⁻ a in s, f a ∂μ ↔ 0 < μ (Function.support f ∩ s) 
goal: 0 < ∫⁻ (a : α) in s, f a ∂μ ↔ 0 < μ (Function.support f ∩ s)

full_name: MeasureTheory.lintegral_iSup_ae
code: /-- Weaker version of the monotone convergence theorem-/
theorem lintegral_iSup_ae {f : ℕ → α → ℝ≥0∞} (hf : ∀ n, Measurable (f n))
    (h_mono : ∀ n, ∀ᵐ a ∂μ, f n a ≤ f n.succ a) : ∫⁻ a, ⨆ n, f n a ∂μ = ⨆ n, ∫⁻ a, f n a ∂μ 
goal: ∫⁻ (a : α), ⨆ n, f n a ∂μ = ⨆ n, ∫⁻ (a : α), f n a ∂μ

full_name: MeasureTheory.lintegral_sub'
code: theorem lintegral_sub' {f g : α → ℝ≥0∞} (hg : AEMeasurable g μ) (hg_fin : ∫⁻ a, g a ∂μ ≠ ∞)
    (h_le : g ≤ᵐ[μ] f) : ∫⁻ a, f a - g a ∂μ = ∫⁻ a, f a ∂μ - ∫⁻ a, g a ∂μ 
goal: ∫⁻ (a : α), f a - g a ∂μ = ∫⁻ (a : α), f a ∂μ - ∫⁻ (a : α), g a ∂μ

full_name: MeasureTheory.lintegral_sub
code: theorem lintegral_sub {f g : α → ℝ≥0∞} (hg : Measurable g) (hg_fin : ∫⁻ a, g a ∂μ ≠ ∞)
    (h_le : g ≤ᵐ[μ] f) : ∫⁻ a, f a - g a ∂μ = ∫⁻ a, f a ∂μ - ∫⁻ a, g a ∂μ 
goal: ∫⁻ (a : α), f a - g a ∂μ = ∫⁻ (a : α), f a ∂μ - ∫⁻ (a : α), g a ∂μ

full_name: MeasureTheory.lintegral_sub_le'
code: theorem lintegral_sub_le' (f g : α → ℝ≥0∞) (hf : AEMeasurable f μ) :
    ∫⁻ x, g x ∂μ - ∫⁻ x, f x ∂μ ≤ ∫⁻ x, g x - f x ∂μ 
goal: ∫⁻ (x : α), g x ∂μ - ∫⁻ (x : α), f x ∂μ ≤ ∫⁻ (x : α), g x - f x ∂μ

full_name: MeasureTheory.lintegral_sub_le
code: theorem lintegral_sub_le (f g : α → ℝ≥0∞) (hf : Measurable f) :
    ∫⁻ x, g x ∂μ - ∫⁻ x, f x ∂μ ≤ ∫⁻ x, g x - f x ∂μ 
goal: ∫⁻ (x : α), g x ∂μ - ∫⁻ (x : α), f x ∂μ ≤ ∫⁻ (x : α), g x - f x ∂μ

full_name: MeasureTheory.lintegral_strict_mono_of_ae_le_of_frequently_ae_lt
code: theorem lintegral_strict_mono_of_ae_le_of_frequently_ae_lt {f g : α → ℝ≥0∞} (hg : AEMeasurable g μ)
    (hfi : ∫⁻ x, f x ∂μ ≠ ∞) (h_le : f ≤ᵐ[μ] g) (h : ∃ᵐ x ∂μ, f x ≠ g x) :
    ∫⁻ x, f x ∂μ < ∫⁻ x, g x ∂μ 
goal: ∫⁻ (x : α), f x ∂μ < ∫⁻ (x : α), g x ∂μ

full_name: MeasureTheory.lintegral_strict_mono_of_ae_le_of_ae_lt_on
code: theorem lintegral_strict_mono_of_ae_le_of_ae_lt_on {f g : α → ℝ≥0∞} (hg : AEMeasurable g μ)
    (hfi : ∫⁻ x, f x ∂μ ≠ ∞) (h_le : f ≤ᵐ[μ] g) {s : Set α} (hμs : μ s ≠ 0)
    (h : ∀ᵐ x ∂μ, x ∈ s → f x < g x) : ∫⁻ x, f x ∂μ < ∫⁻ x, g x ∂μ 
goal: ∫⁻ (x : α), f x ∂μ < ∫⁻ (x : α), g x ∂μ

full_name: MeasureTheory.lintegral_strict_mono
code: theorem lintegral_strict_mono {f g : α → ℝ≥0∞} (hμ : μ ≠ 0) (hg : AEMeasurable g μ)
    (hfi : ∫⁻ x, f x ∂μ ≠ ∞) (h : ∀ᵐ x ∂μ, f x < g x) : ∫⁻ x, f x ∂μ < ∫⁻ x, g x ∂μ 
goal: ∫⁻ (x : α), f x ∂μ < ∫⁻ (x : α), g x ∂μ

full_name: MeasureTheory.setLIntegral_strict_mono
code: theorem setLIntegral_strict_mono {f g : α → ℝ≥0∞} {s : Set α} (hsm : MeasurableSet s)
    (hs : μ s ≠ 0) (hg : Measurable g) (hfi : ∫⁻ x in s, f x ∂μ ≠ ∞)
    (h : ∀ᵐ x ∂μ, x ∈ s → f x < g x) : ∫⁻ x in s, f x ∂μ < ∫⁻ x in s, g x ∂μ 
goal: ∫⁻ (x : α) in s, f x ∂μ < ∫⁻ (x : α) in s, g x ∂μ

full_name: MeasureTheory.lintegral_iInf_ae
code: /-- Monotone convergence theorem for nonincreasing sequences of functions -/
theorem lintegral_iInf_ae {f : ℕ → α → ℝ≥0∞} (h_meas : ∀ n, Measurable (f n))
    (h_mono : ∀ n : ℕ, f n.succ ≤ᵐ[μ] f n) (h_fin : ∫⁻ a, f 0 a ∂μ ≠ ∞) :
    ∫⁻ a, ⨅ n, f n a ∂μ = ⨅ n, ∫⁻ a, f n a ∂μ 
goal: ∫⁻ (a : α), ⨅ n, f n a ∂μ = ⨅ n, ∫⁻ (a : α), f n a ∂μ

full_name: MeasureTheory.lintegral_iInf
code: /-- Monotone convergence theorem for nonincreasing sequences of functions -/
theorem lintegral_iInf {f : ℕ → α → ℝ≥0∞} (h_meas : ∀ n, Measurable (f n)) (h_anti : Antitone f)
    (h_fin : ∫⁻ a, f 0 a ∂μ ≠ ∞) : ∫⁻ a, ⨅ n, f n a ∂μ = ⨅ n, ∫⁻ a, f n a ∂μ 
goal: ∫⁻ (a : α), ⨅ n, f n a ∂μ = ⨅ n, ∫⁻ (a : α), f n a ∂μ

full_name: MeasureTheory.lintegral_iInf'
code: theorem lintegral_iInf' {f : ℕ → α → ℝ≥0∞} (h_meas : ∀ n, AEMeasurable (f n) μ)
    (h_anti : ∀ᵐ a ∂μ, Antitone (fun i ↦ f i a)) (h_fin : ∫⁻ a, f 0 a ∂μ ≠ ∞) :
    ∫⁻ a, ⨅ n, f n a ∂μ = ⨅ n, ∫⁻ a, f n a ∂μ 
goal: ∫⁻ (a : α), ⨅ n, f n a ∂μ = ⨅ n, ∫⁻ (a : α), f n a ∂μ

full_name: MeasureTheory.lintegral_iInf_directed_of_measurable
code: /-- Monotone convergence for an infimum over a directed family and indexed by a countable type -/
theorem lintegral_iInf_directed_of_measurable {mα : MeasurableSpace α} [Countable β]
    {f : β → α → ℝ≥0∞} {μ : Measure α} (hμ : μ ≠ 0) (hf : ∀ b, Measurable (f b))
    (hf_int : ∀ b, ∫⁻ a, f b a ∂μ ≠ ∞) (h_directed : Directed (· ≥ ·) f) :
    ∫⁻ a, ⨅ b, f b a ∂μ = ⨅ b, ∫⁻ a, f b a ∂μ 
goal: ∫⁻ (a : α), ⨅ b, f b a ∂μ = ⨅ b, ∫⁻ (a : α), f b a ∂μ

full_name: MeasureTheory.lintegral_liminf_le'
code: /-- Known as Fatou's lemma, version with `AEMeasurable` functions -/
theorem lintegral_liminf_le' {f : ℕ → α → ℝ≥0∞} (h_meas : ∀ n, AEMeasurable (f n) μ) :
    ∫⁻ a, liminf (fun n => f n a) atTop ∂μ ≤ liminf (fun n => ∫⁻ a, f n a ∂μ) atTop 
goal: ∫⁻ (a : α), Filter.liminf (fun n => f n a) Filter.atTop ∂μ ≤ Filter.liminf (fun n => ∫⁻ (a : α), f n a ∂μ) Filter.atTop

full_name: MeasureTheory.lintegral_liminf_le
code: /-- Known as Fatou's lemma -/
theorem lintegral_liminf_le {f : ℕ → α → ℝ≥0∞} (h_meas : ∀ n, Measurable (f n)) :
    ∫⁻ a, liminf (fun n => f n a) atTop ∂μ ≤ liminf (fun n => ∫⁻ a, f n a ∂μ) atTop 
goal: ∫⁻ (a : α), Filter.liminf (fun n => f n a) Filter.atTop ∂μ ≤ Filter.liminf (fun n => ∫⁻ (a : α), f n a ∂μ) Filter.atTop

full_name: MeasureTheory.limsup_lintegral_le
code: theorem limsup_lintegral_le {f : ℕ → α → ℝ≥0∞} (g : α → ℝ≥0∞) (hf_meas : ∀ n, Measurable (f n))
    (h_bound : ∀ n, f n ≤ᵐ[μ] g) (h_fin : ∫⁻ a, g a ∂μ ≠ ∞) :
    limsup (fun n => ∫⁻ a, f n a ∂μ) atTop ≤ ∫⁻ a, limsup (fun n => f n a) atTop ∂μ 
goal: Filter.limsup (fun n => ∫⁻ (a : α), f n a ∂μ) Filter.atTop ≤ ∫⁻ (a : α), Filter.limsup (fun n => f n a) Filter.atTop ∂μ

full_name: MeasureTheory.tendsto_lintegral_of_dominated_convergence
code: /-- Dominated convergence theorem for nonnegative functions -/
theorem tendsto_lintegral_of_dominated_convergence {F : ℕ → α → ℝ≥0∞} {f : α → ℝ≥0∞}
    (bound : α → ℝ≥0∞) (hF_meas : ∀ n, Measurable (F n)) (h_bound : ∀ n, F n ≤ᵐ[μ] bound)
    (h_fin : ∫⁻ a, bound a ∂μ ≠ ∞) (h_lim : ∀ᵐ a ∂μ, Tendsto (fun n => F n a) atTop (𝓝 (f a))) :
    Tendsto (fun n => ∫⁻ a, F n a ∂μ) atTop (𝓝 (∫⁻ a, f a ∂μ)) 
goal: Filter.Tendsto (fun n => ∫⁻ (a : α), F n a ∂μ) Filter.atTop (𝓝 (∫⁻ (a : α), f a ∂μ))

full_name: MeasureTheory.tendsto_lintegral_of_dominated_convergence'
code: /-- Dominated convergence theorem for nonnegative functions which are just almost everywhere
measurable. -/
theorem tendsto_lintegral_of_dominated_convergence' {F : ℕ → α → ℝ≥0∞} {f : α → ℝ≥0∞}
    (bound : α → ℝ≥0∞) (hF_meas : ∀ n, AEMeasurable (F n) μ) (h_bound : ∀ n, F n ≤ᵐ[μ] bound)
    (h_fin : ∫⁻ a, bound a ∂μ ≠ ∞) (h_lim : ∀ᵐ a ∂μ, Tendsto (fun n => F n a) atTop (𝓝 (f a))) :
    Tendsto (fun n => ∫⁻ a, F n a ∂μ) atTop (𝓝 (∫⁻ a, f a ∂μ)) 
goal: Filter.Tendsto (fun n => ∫⁻ (a : α), F n a ∂μ) Filter.atTop (𝓝 (∫⁻ (a : α), f a ∂μ))

full_name: MeasureTheory.tendsto_lintegral_filter_of_dominated_convergence
code: /-- Dominated convergence theorem for filters with a countable basis -/
theorem tendsto_lintegral_filter_of_dominated_convergence {ι} {l : Filter ι}
    [l.IsCountablyGenerated] {F : ι → α → ℝ≥0∞} {f : α → ℝ≥0∞} (bound : α → ℝ≥0∞)
    (hF_meas : ∀ᶠ n in l, Measurable (F n)) (h_bound : ∀ᶠ n in l, ∀ᵐ a ∂μ, F n a ≤ bound a)
    (h_fin : ∫⁻ a, bound a ∂μ ≠ ∞) (h_lim : ∀ᵐ a ∂μ, Tendsto (fun n => F n a) l (𝓝 (f a))) :
    Tendsto (fun n => ∫⁻ a, F n a ∂μ) l (𝓝 <| ∫⁻ a, f a ∂μ) 
goal: Filter.Tendsto (fun n => ∫⁻ (a : α), F n a ∂μ) l (𝓝 (∫⁻ (a : α), f a ∂μ))

full_name: MeasureTheory.lintegral_tendsto_of_tendsto_of_antitone
code: theorem lintegral_tendsto_of_tendsto_of_antitone {f : ℕ → α → ℝ≥0∞} {F : α → ℝ≥0∞}
    (hf : ∀ n, AEMeasurable (f n) μ) (h_anti : ∀ᵐ x ∂μ, Antitone fun n ↦ f n x)
    (h0 : ∫⁻ a, f 0 a ∂μ ≠ ∞)
    (h_tendsto : ∀ᵐ x ∂μ, Tendsto (fun n ↦ f n x) atTop (𝓝 (F x))) :
    Tendsto (fun n ↦ ∫⁻ x, f n x ∂μ) atTop (𝓝 (∫⁻ x, F x ∂μ)) 
goal: Filter.Tendsto (fun n => ∫⁻ (x : α), f n x ∂μ) Filter.atTop (𝓝 (∫⁻ (x : α), F x ∂μ))

full_name: MeasureTheory.lintegral_iSup_directed_of_measurable
code: /-- Monotone convergence for a supremum over a directed family and indexed by a countable type -/
theorem lintegral_iSup_directed_of_measurable [Countable β] {f : β → α → ℝ≥0∞}
    (hf : ∀ b, Measurable (f b)) (h_directed : Directed (· ≤ ·) f) :
    ∫⁻ a, ⨆ b, f b a ∂μ = ⨆ b, ∫⁻ a, f b a ∂μ 
goal: ∫⁻ (a : α), ⨆ b, f b a ∂μ = ⨆ b, ∫⁻ (a : α), f b a ∂μ

full_name: MeasureTheory.lintegral_iSup_directed
code: /-- Monotone convergence for a supremum over a directed family and indexed by a countable type. -/
theorem lintegral_iSup_directed [Countable β] {f : β → α → ℝ≥0∞} (hf : ∀ b, AEMeasurable (f b) μ)
    (h_directed : Directed (· ≤ ·) f) : ∫⁻ a, ⨆ b, f b a ∂μ = ⨆ b, ∫⁻ a, f b a ∂μ 
goal: ∫⁻ (a : α), ⨆ b, f b a ∂μ = ⨆ b, ∫⁻ (a : α), f b a ∂μ

full_name: MeasureTheory.lintegral_tsum
code: theorem lintegral_tsum [Countable β] {f : β → α → ℝ≥0∞} (hf : ∀ i, AEMeasurable (f i) μ) :
    ∫⁻ a, ∑' i, f i a ∂μ = ∑' i, ∫⁻ a, f i a ∂μ 
goal: ∫⁻ (a : α), ∑' (i : β), f i a ∂μ = ∑' (i : β), ∫⁻ (a : α), f i a ∂μ

full_name: MeasureTheory.lintegral_iUnion₀
code: theorem lintegral_iUnion₀ [Countable β] {s : β → Set α} (hm : ∀ i, NullMeasurableSet (s i) μ)
    (hd : Pairwise (AEDisjoint μ on s)) (f : α → ℝ≥0∞) :
    ∫⁻ a in ⋃ i, s i, f a ∂μ = ∑' i, ∫⁻ a in s i, f a ∂μ 
goal: ∫⁻ (a : α) in ⋃ i, s i, f a ∂μ = ∑' (i : β), ∫⁻ (a : α) in s i, f a ∂μ

full_name: MeasureTheory.lintegral_iUnion
code: theorem lintegral_iUnion [Countable β] {s : β → Set α} (hm : ∀ i, MeasurableSet (s i))
    (hd : Pairwise (Disjoint on s)) (f : α → ℝ≥0∞) :
    ∫⁻ a in ⋃ i, s i, f a ∂μ = ∑' i, ∫⁻ a in s i, f a ∂μ 
goal: ∫⁻ (a : α) in ⋃ i, s i, f a ∂μ = ∑' (i : β), ∫⁻ (a : α) in s i, f a ∂μ

full_name: MeasureTheory.lintegral_biUnion₀
code: theorem lintegral_biUnion₀ {t : Set β} {s : β → Set α} (ht : t.Countable)
    (hm : ∀ i ∈ t, NullMeasurableSet (s i) μ) (hd : t.Pairwise (AEDisjoint μ on s)) (f : α → ℝ≥0∞) :
    ∫⁻ a in ⋃ i ∈ t, s i, f a ∂μ = ∑' i : t, ∫⁻ a in s i, f a ∂μ 
goal: ∫⁻ (a : α) in ⋃ i ∈ t, s i, f a ∂μ = ∑' (i : ↑t), ∫⁻ (a : α) in s ↑i, f a ∂μ

full_name: MeasureTheory.lintegral_biUnion
code: theorem lintegral_biUnion {t : Set β} {s : β → Set α} (ht : t.Countable)
    (hm : ∀ i ∈ t, MeasurableSet (s i)) (hd : t.PairwiseDisjoint s) (f : α → ℝ≥0∞) :
    ∫⁻ a in ⋃ i ∈ t, s i, f a ∂μ = ∑' i : t, ∫⁻ a in s i, f a ∂μ 
goal: ∫⁻ (a : α) in ⋃ i ∈ t, s i, f a ∂μ = ∑' (i : ↑t), ∫⁻ (a : α) in s ↑i, f a ∂μ

full_name: MeasureTheory.lintegral_biUnion_finset₀
code: theorem lintegral_biUnion_finset₀ {s : Finset β} {t : β → Set α}
    (hd : Set.Pairwise (↑s) (AEDisjoint μ on t)) (hm : ∀ b ∈ s, NullMeasurableSet (t b) μ)
    (f : α → ℝ≥0∞) : ∫⁻ a in ⋃ b ∈ s, t b, f a ∂μ = ∑ b ∈ s, ∫⁻ a in t b, f a ∂μ 
goal: ∫⁻ (a : α) in ⋃ b ∈ s, t b, f a ∂μ = ∑ b ∈ s, ∫⁻ (a : α) in t b, f a ∂μ

full_name: MeasureTheory.lintegral_biUnion_finset
code: theorem lintegral_biUnion_finset {s : Finset β} {t : β → Set α} (hd : Set.PairwiseDisjoint (↑s) t)
    (hm : ∀ b ∈ s, MeasurableSet (t b)) (f : α → ℝ≥0∞) :
    ∫⁻ a in ⋃ b ∈ s, t b, f a ∂μ = ∑ b ∈ s, ∫⁻ a in t b, f a ∂μ 
goal: ∫⁻ (a : α) in ⋃ b ∈ s, t b, f a ∂μ = ∑ b ∈ s, ∫⁻ (a : α) in t b, f a ∂μ

full_name: MeasureTheory.lintegral_iUnion_le
code: theorem lintegral_iUnion_le [Countable β] (s : β → Set α) (f : α → ℝ≥0∞) :
    ∫⁻ a in ⋃ i, s i, f a ∂μ ≤ ∑' i, ∫⁻ a in s i, f a ∂μ 
goal: ∫⁻ (a : α) in ⋃ i, s i, f a ∂μ ≤ ∑' (i : β), ∫⁻ (a : α) in s i, f a ∂μ

full_name: MeasureTheory.lintegral_union
code: theorem lintegral_union {f : α → ℝ≥0∞} {A B : Set α} (hB : MeasurableSet B) (hAB : Disjoint A B) :
    ∫⁻ a in A ∪ B, f a ∂μ = ∫⁻ a in A, f a ∂μ + ∫⁻ a in B, f a ∂μ 
goal: ∫⁻ (a : α) in A ∪ B, f a ∂μ = ∫⁻ (a : α) in A, f a ∂μ + ∫⁻ (a : α) in B, f a ∂μ

full_name: MeasureTheory.lintegral_union_le
code: theorem lintegral_union_le (f : α → ℝ≥0∞) (s t : Set α) :
    ∫⁻ a in s ∪ t, f a ∂μ ≤ ∫⁻ a in s, f a ∂μ + ∫⁻ a in t, f a ∂μ 
goal: ∫⁻ (a : α) in s ∪ t, f a ∂μ ≤ ∫⁻ (a : α) in s, f a ∂μ + ∫⁻ (a : α) in t, f a ∂μ

full_name: MeasureTheory.lintegral_inter_add_diff
code: theorem lintegral_inter_add_diff {B : Set α} (f : α → ℝ≥0∞) (A : Set α) (hB : MeasurableSet B) :
    ∫⁻ x in A ∩ B, f x ∂μ + ∫⁻ x in A \ B, f x ∂μ = ∫⁻ x in A, f x ∂μ 
goal: ∫⁻ (x : α) in A ∩ B, f x ∂μ + ∫⁻ (x : α) in A \ B, f x ∂μ = ∫⁻ (x : α) in A, f x ∂μ

full_name: MeasureTheory.lintegral_add_compl
code: theorem lintegral_add_compl (f : α → ℝ≥0∞) {A : Set α} (hA : MeasurableSet A) :
    ∫⁻ x in A, f x ∂μ + ∫⁻ x in Aᶜ, f x ∂μ = ∫⁻ x, f x ∂μ 
goal: ∫⁻ (x : α) in A, f x ∂μ + ∫⁻ (x : α) in Aᶜ, f x ∂μ = ∫⁻ (x : α), f x ∂μ

full_name: MeasureTheory.setLintegral_compl
code: theorem setLintegral_compl {f : α → ℝ≥0∞} {s : Set α} (hsm : MeasurableSet s)
    (hfs : ∫⁻ x in s, f x ∂μ ≠ ∞) :
    ∫⁻ x in sᶜ, f x ∂μ = ∫⁻ x, f x ∂μ - ∫⁻ x in s, f x ∂μ 
goal: ∫⁻ (x : α) in sᶜ, f x ∂μ = ∫⁻ (x : α), f x ∂μ - ∫⁻ (x : α) in s, f x ∂μ

full_name: MeasureTheory.setLIntegral_iUnion_of_directed
code: theorem setLIntegral_iUnion_of_directed {ι : Type*} [Countable ι]
    (f : α → ℝ≥0∞) {s : ι → Set α} (hd : Directed (· ⊆ ·) s) :
    ∫⁻ x in ⋃ i, s i, f x ∂μ = ⨆ i, ∫⁻ x in s i, f x ∂μ 
goal: ∫⁻ (x : α) in ⋃ i, s i, f x ∂μ = ⨆ i, ∫⁻ (x : α) in s i, f x ∂μ

full_name: MeasureTheory.lintegral_max
code: theorem lintegral_max {f g : α → ℝ≥0∞} (hf : Measurable f) (hg : Measurable g) :
    ∫⁻ x, max (f x) (g x) ∂μ =
      ∫⁻ x in { x | f x ≤ g x }, g x ∂μ + ∫⁻ x in { x | g x < f x }, f x ∂μ 
goal: ∫⁻ (x : α), max (f x) (g x) ∂μ = ∫⁻ (x : α) in {x | f x ≤ g x}, g x ∂μ + ∫⁻ (x : α) in {x | g x < f x}, f x ∂μ

full_name: MeasureTheory.setLIntegral_max
code: theorem setLIntegral_max {f g : α → ℝ≥0∞} (hf : Measurable f) (hg : Measurable g) (s : Set α) :
    ∫⁻ x in s, max (f x) (g x) ∂μ =
      ∫⁻ x in s ∩ { x | f x ≤ g x }, g x ∂μ + ∫⁻ x in s ∩ { x | g x < f x }, f x ∂μ 
goal: ∫⁻ (x : α) in s, max (f x) (g x) ∂μ =
  ∫⁻ (x : α) in s ∩ {x | f x ≤ g x}, g x ∂μ + ∫⁻ (x : α) in s ∩ {x | g x < f x}, f x ∂μ

full_name: MeasureTheory.lintegral_map
code: theorem lintegral_map {mβ : MeasurableSpace β} {f : β → ℝ≥0∞} {g : α → β} (hf : Measurable f)
    (hg : Measurable g) : ∫⁻ a, f a ∂map g μ = ∫⁻ a, f (g a) ∂μ 
goal: ∫⁻ (a : β), f a ∂MeasureTheory.Measure.map g μ = ∫⁻ (a : α), f (g a) ∂μ

full_name: MeasureTheory.lintegral_map'
code: theorem lintegral_map' {mβ : MeasurableSpace β} {f : β → ℝ≥0∞} {g : α → β}
    (hf : AEMeasurable f (Measure.map g μ)) (hg : AEMeasurable g μ) :
    ∫⁻ a, f a ∂Measure.map g μ = ∫⁻ a, f (g a) ∂μ 
goal: ∫⁻ (a : β), f a ∂MeasureTheory.Measure.map g μ = ∫⁻ (a : α), f (g a) ∂μ

full_name: MeasureTheory.lintegral_map_le
code: theorem lintegral_map_le {mβ : MeasurableSpace β} (f : β → ℝ≥0∞) {g : α → β} (hg : Measurable g) :
    ∫⁻ a, f a ∂Measure.map g μ ≤ ∫⁻ a, f (g a) ∂μ 
goal: ∫⁻ (a : β), f a ∂MeasureTheory.Measure.map g μ ≤ ∫⁻ (a : α), f (g a) ∂μ

full_name: MeasureTheory.lintegral_comp
code: theorem lintegral_comp [MeasurableSpace β] {f : β → ℝ≥0∞} {g : α → β} (hf : Measurable f)
    (hg : Measurable g) : lintegral μ (f ∘ g) = ∫⁻ a, f a ∂map g μ 
goal: MeasureTheory.lintegral μ (f ∘ g) = ∫⁻ (a : β), f a ∂MeasureTheory.Measure.map g μ

full_name: MeasureTheory.setLIntegral_map
code: theorem setLIntegral_map [MeasurableSpace β] {f : β → ℝ≥0∞} {g : α → β} {s : Set β}
    (hs : MeasurableSet s) (hf : Measurable f) (hg : Measurable g) :
    ∫⁻ y in s, f y ∂map g μ = ∫⁻ x in g ⁻¹' s, f (g x) ∂μ 
goal: ∫⁻ (y : β) in s, f y ∂MeasureTheory.Measure.map g μ = ∫⁻ (x : α) in g ⁻¹' s, f (g x) ∂μ

full_name: MeasureTheory.lintegral_indicator_const_comp
code: theorem lintegral_indicator_const_comp {mβ : MeasurableSpace β} {f : α → β} {s : Set β}
    (hf : Measurable f) (hs : MeasurableSet s) (c : ℝ≥0∞) :
    ∫⁻ a, s.indicator (fun _ => c) (f a) ∂μ = c * μ (f ⁻¹' s) 
goal: ∫⁻ (a : α), s.indicator (fun x => c) (f a) ∂μ = c * μ (f ⁻¹' s)

full_name: MeasurableEmbedding.lintegral_map
code: /-- If `g : α → β` is a measurable embedding and `f : β → ℝ≥0∞` is any function (not necessarily
measurable), then `∫⁻ a, f a ∂(map g μ) = ∫⁻ a, f (g a) ∂μ`. Compare with `lintegral_map` which
applies to any measurable `g : α → β` but requires that `f` is measurable as well. -/
theorem _root_.MeasurableEmbedding.lintegral_map [MeasurableSpace β] {g : α → β}
    (hg : MeasurableEmbedding g) (f : β → ℝ≥0∞) : ∫⁻ a, f a ∂map g μ = ∫⁻ a, f (g a) ∂μ 
goal: ∫⁻ (a : β), f a ∂MeasureTheory.Measure.map g μ = ∫⁻ (a : α), f (g a) ∂μ

full_name: MeasureTheory.lintegral_map_equiv
code: /-- The `lintegral` transforms appropriately under a measurable equivalence `g : α ≃ᵐ β`.
(Compare `lintegral_map`, which applies to a wider class of functions `g : α → β`, but requires
measurability of the function being integrated.) -/
theorem lintegral_map_equiv [MeasurableSpace β] (f : β → ℝ≥0∞) (g : α ≃ᵐ β) :
    ∫⁻ a, f a ∂map g μ = ∫⁻ a, f (g a) ∂μ 
goal: ∫⁻ (a : β), f a ∂MeasureTheory.Measure.map (⇑g) μ = ∫⁻ (a : α), f (g a) ∂μ

full_name: MeasureTheory.MeasurePreserving.lintegral_map_equiv
code: protected theorem MeasurePreserving.lintegral_map_equiv [MeasurableSpace β] {ν : Measure β}
    (f : β → ℝ≥0∞) (g : α ≃ᵐ β) (hg : MeasurePreserving g μ ν) :
    ∫⁻ a, f a ∂ν = ∫⁻ a, f (g a) ∂μ 
goal: ∫⁻ (a : β), f a ∂ν = ∫⁻ (a : α), f (g a) ∂μ

full_name: MeasureTheory.MeasurePreserving.lintegral_comp
code: theorem MeasurePreserving.lintegral_comp {mb : MeasurableSpace β} {ν : Measure β} {g : α → β}
    (hg : MeasurePreserving g μ ν) {f : β → ℝ≥0∞} (hf : Measurable f) :
    ∫⁻ a, f (g a) ∂μ = ∫⁻ b, f b ∂ν 
goal: ∫⁻ (a : α), f (g a) ∂μ = ∫⁻ (b : β), f b ∂ν

full_name: MeasureTheory.MeasurePreserving.lintegral_comp_emb
code: theorem MeasurePreserving.lintegral_comp_emb {mb : MeasurableSpace β} {ν : Measure β} {g : α → β}
    (hg : MeasurePreserving g μ ν) (hge : MeasurableEmbedding g) (f : β → ℝ≥0∞) :
    ∫⁻ a, f (g a) ∂μ = ∫⁻ b, f b ∂ν 
goal: ∫⁻ (a : α), f (g a) ∂μ = ∫⁻ (b : β), f b ∂ν

full_name: MeasureTheory.MeasurePreserving.setLIntegral_comp_preimage
code: theorem MeasurePreserving.setLIntegral_comp_preimage {mb : MeasurableSpace β} {ν : Measure β}
    {g : α → β} (hg : MeasurePreserving g μ ν) {s : Set β} (hs : MeasurableSet s) {f : β → ℝ≥0∞}
    (hf : Measurable f) : ∫⁻ a in g ⁻¹' s, f (g a) ∂μ = ∫⁻ b in s, f b ∂ν 
goal: ∫⁻ (a : α) in g ⁻¹' s, f (g a) ∂μ = ∫⁻ (b : β) in s, f b ∂ν

full_name: MeasureTheory.MeasurePreserving.setLIntegral_comp_preimage_emb
code: theorem MeasurePreserving.setLIntegral_comp_preimage_emb {mb : MeasurableSpace β} {ν : Measure β}
    {g : α → β} (hg : MeasurePreserving g μ ν) (hge : MeasurableEmbedding g) (f : β → ℝ≥0∞)
    (s : Set β) : ∫⁻ a in g ⁻¹' s, f (g a) ∂μ = ∫⁻ b in s, f b ∂ν 
goal: ∫⁻ (a : α) in g ⁻¹' s, f (g a) ∂μ = ∫⁻ (b : β) in s, f b ∂ν

full_name: MeasureTheory.MeasurePreserving.setLIntegral_comp_emb
code: theorem MeasurePreserving.setLIntegral_comp_emb {mb : MeasurableSpace β} {ν : Measure β}
    {g : α → β} (hg : MeasurePreserving g μ ν) (hge : MeasurableEmbedding g) (f : β → ℝ≥0∞)
    (s : Set α) : ∫⁻ a in s, f (g a) ∂μ = ∫⁻ b in g '' s, f b ∂ν 
goal: ∫⁻ (a : α) in s, f (g a) ∂μ = ∫⁻ (b : β) in g '' s, f b ∂ν

full_name: MeasureTheory.lintegral_subtype_comap
code: theorem lintegral_subtype_comap {s : Set α} (hs : MeasurableSet s) (f : α → ℝ≥0∞) :
    ∫⁻ x : s, f x ∂(μ.comap (↑)) = ∫⁻ x in s, f x ∂μ 
goal: ∫⁻ (x : ↑s), f ↑x ∂MeasureTheory.Measure.comap Subtype.val μ = ∫⁻ (x : α) in s, f x ∂μ

full_name: MeasureTheory.setLIntegral_subtype
code: theorem setLIntegral_subtype {s : Set α} (hs : MeasurableSet s) (t : Set s) (f : α → ℝ≥0∞) :
    ∫⁻ x in t, f x ∂(μ.comap (↑)) = ∫⁻ x in (↑) '' t, f x ∂μ 
goal: ∫⁻ (x : ↑s) in t, f ↑x ∂MeasureTheory.Measure.comap Subtype.val μ = ∫⁻ (x : α) in Subtype.val '' t, f x ∂μ

full_name: MeasureTheory.exists_setLintegral_compl_lt
code: /-- If `f : α → ℝ≥0∞` has finite integral, then there exists a measurable set `s` of finite measure
such that the integral of `f` over `sᶜ` is less than a given positive number.

Also used to prove an `Lᵖ`-norm version in
`MeasureTheory.Memℒp.exists_eLpNorm_indicator_compl_le`. -/
theorem exists_setLintegral_compl_lt {f : α → ℝ≥0∞} (hf : ∫⁻ a, f a ∂μ ≠ ∞)
    {ε : ℝ≥0∞} (hε : ε ≠ 0) :
    ∃ s : Set α, MeasurableSet s ∧ μ s < ∞ ∧ ∫⁻ a in sᶜ, f a ∂μ < ε 
goal: ∃ s, MeasurableSet s ∧ μ s < ⊤ ∧ ∫⁻ (a : α) in sᶜ, f a ∂μ < ε

full_name: MeasureTheory.exists_measurable_le_setLintegral_eq_of_integrable
code: /-- For any function `f : α → ℝ≥0∞`, there exists a measurable function `g ≤ f` with the same
integral over any measurable set. -/
theorem exists_measurable_le_setLintegral_eq_of_integrable {f : α → ℝ≥0∞} (hf : ∫⁻ a, f a ∂μ ≠ ∞) :
    ∃ (g : α → ℝ≥0∞), Measurable g ∧ g ≤ f ∧ ∀ s : Set α, MeasurableSet s →
      ∫⁻ a in s, f a ∂μ = ∫⁻ a in s, g a ∂μ 
goal: ∃ g, Measurable g ∧ g ≤ f ∧ ∀ (s : Set α), MeasurableSet s → ∫⁻ (a : α) in s, f a ∂μ = ∫⁻ (a : α) in s, g a ∂μ

full_name: MeasureTheory.lintegral_dirac'
code: theorem lintegral_dirac' (a : α) {f : α → ℝ≥0∞} (hf : Measurable f) : ∫⁻ a, f a ∂dirac a = f a 
goal: ∫⁻ (a : α), f a ∂MeasureTheory.Measure.dirac a = f a

full_name: MeasureTheory.lintegral_dirac
code: theorem lintegral_dirac [MeasurableSingletonClass α] (a : α) (f : α → ℝ≥0∞) :
    ∫⁻ a, f a ∂dirac a = f a 
goal: ∫⁻ (a : α), f a ∂MeasureTheory.Measure.dirac a = f a

full_name: MeasureTheory.setLIntegral_dirac'
code: theorem setLIntegral_dirac' {a : α} {f : α → ℝ≥0∞} (hf : Measurable f) {s : Set α}
    (hs : MeasurableSet s) [Decidable (a ∈ s)] :
    ∫⁻ x in s, f x ∂Measure.dirac a = if a ∈ s then f a else 0 
goal: ∫⁻ (x : α) in s, f x ∂MeasureTheory.Measure.dirac a = if a ∈ s then f a else 0

full_name: MeasureTheory.setLIntegral_dirac
code: theorem setLIntegral_dirac {a : α} (f : α → ℝ≥0∞) (s : Set α) [MeasurableSingletonClass α]
    [Decidable (a ∈ s)] : ∫⁻ x in s, f x ∂Measure.dirac a = if a ∈ s then f a else 0 
goal: ∫⁻ (x : α) in s, f x ∂MeasureTheory.Measure.dirac a = if a ∈ s then f a else 0

full_name: MeasureTheory.lintegral_count'
code: theorem lintegral_count' {f : α → ℝ≥0∞} (hf : Measurable f) : ∫⁻ a, f a ∂count = ∑' a, f a 
goal: ∫⁻ (a : α), f a ∂MeasureTheory.Measure.count = ∑' (a : α), f a

full_name: MeasureTheory.lintegral_count
code: theorem lintegral_count [MeasurableSingletonClass α] (f : α → ℝ≥0∞) :
    ∫⁻ a, f a ∂count = ∑' a, f a 
goal: ∫⁻ (a : α), f a ∂MeasureTheory.Measure.count = ∑' (a : α), f a

full_name: ENNReal.tsum_const_eq
code: theorem _root_.ENNReal.tsum_const_eq [MeasurableSingletonClass α] (c : ℝ≥0∞) :
    ∑' _ : α, c = c * Measure.count (univ : Set α) 
goal: ∑' (x : α), c = c * MeasureTheory.Measure.count Set.univ

full_name: ENNReal.count_const_le_le_of_tsum_le
code: /-- Markov's inequality for the counting measure with hypothesis using `tsum` in `ℝ≥0∞`. -/
theorem _root_.ENNReal.count_const_le_le_of_tsum_le [MeasurableSingletonClass α] {a : α → ℝ≥0∞}
    (a_mble : Measurable a) {c : ℝ≥0∞} (tsum_le_c : ∑' i, a i ≤ c) {ε : ℝ≥0∞} (ε_ne_zero : ε ≠ 0)
    (ε_ne_top : ε ≠ ∞) : Measure.count { i : α | ε ≤ a i } ≤ c / ε 
goal: MeasureTheory.Measure.count {i | ε ≤ a i} ≤ c / ε

full_name: NNReal.count_const_le_le_of_tsum_le
code: /-- Markov's inequality for counting measure with hypothesis using `tsum` in `ℝ≥0`. -/
theorem _root_.NNReal.count_const_le_le_of_tsum_le [MeasurableSingletonClass α] {a : α → ℝ≥0}
    (a_mble : Measurable a) (a_summable : Summable a) {c : ℝ≥0} (tsum_le_c : ∑' i, a i ≤ c)
    {ε : ℝ≥0} (ε_ne_zero : ε ≠ 0) : Measure.count { i : α | ε ≤ a i } ≤ c / ε 
goal: MeasureTheory.Measure.count {i | ε ≤ a i} ≤ ↑c / ↑ε

full_name: MeasureTheory.lintegral_countable'
code: theorem lintegral_countable' [Countable α] [MeasurableSingletonClass α] (f : α → ℝ≥0∞) :
    ∫⁻ a, f a ∂μ = ∑' a, f a * μ {a} 
goal: ∫⁻ (a : α), f a ∂μ = ∑' (a : α), f a * μ {a}

full_name: MeasureTheory.lintegral_singleton'
code: theorem lintegral_singleton' {f : α → ℝ≥0∞} (hf : Measurable f) (a : α) :
    ∫⁻ x in {a}, f x ∂μ = f a * μ {a} 
goal: ∫⁻ (x : α) in {a}, f x ∂μ = f a * μ {a}

full_name: MeasureTheory.lintegral_singleton
code: theorem lintegral_singleton [MeasurableSingletonClass α] (f : α → ℝ≥0∞) (a : α) :
    ∫⁻ x in {a}, f x ∂μ = f a * μ {a} 
goal: ∫⁻ (x : α) in {a}, f x ∂μ = f a * μ {a}

full_name: MeasureTheory.lintegral_countable
code: theorem lintegral_countable [MeasurableSingletonClass α] (f : α → ℝ≥0∞) {s : Set α}
    (hs : s.Countable) : ∫⁻ a in s, f a ∂μ = ∑' a : s, f a * μ {(a : α)} 
goal: ∫⁻ (a : α) in s, f a ∂μ = ∑' (a : ↑s), f ↑a * μ {↑a}

full_name: MeasureTheory.lintegral_insert
code: theorem lintegral_insert [MeasurableSingletonClass α] {a : α} {s : Set α} (h : a ∉ s)
    (f : α → ℝ≥0∞) : ∫⁻ x in insert a s, f x ∂μ = f a * μ {a} + ∫⁻ x in s, f x ∂μ 
goal: ∫⁻ (x : α) in insert a s, f x ∂μ = f a * μ {a} + ∫⁻ (x : α) in s, f x ∂μ

full_name: MeasureTheory.lintegral_finset
code: theorem lintegral_finset [MeasurableSingletonClass α] (s : Finset α) (f : α → ℝ≥0∞) :
    ∫⁻ x in s, f x ∂μ = ∑ x ∈ s, f x * μ {x} 
goal: ∫⁻ (x : α) in ↑s, f x ∂μ = ∑ x ∈ s, f x * μ {x}

full_name: MeasureTheory.lintegral_fintype
code: theorem lintegral_fintype [MeasurableSingletonClass α] [Fintype α] (f : α → ℝ≥0∞) :
    ∫⁻ x, f x ∂μ = ∑ x, f x * μ {x} 
goal: ∫⁻ (x : α), f x ∂μ = ∑ x : α, f x * μ {x}

full_name: MeasureTheory.lintegral_unique
code: theorem lintegral_unique [Unique α] (f : α → ℝ≥0∞) : ∫⁻ x, f x ∂μ = f default * μ univ 
goal: ∫⁻ (x : α), f x ∂μ = f default * μ Set.univ

full_name: MeasureTheory.ae_lt_top'
code: theorem ae_lt_top' {f : α → ℝ≥0∞} (hf : AEMeasurable f μ) (h2f : ∫⁻ x, f x ∂μ ≠ ∞) :
    ∀ᵐ x ∂μ, f x < ∞ 
goal: ∀ᵐ (x : α) ∂μ, f x < ⊤

full_name: MeasureTheory.ae_lt_top
code: theorem ae_lt_top {f : α → ℝ≥0∞} (hf : Measurable f) (h2f : ∫⁻ x, f x ∂μ ≠ ∞) :
    ∀ᵐ x ∂μ, f x < ∞ 
goal: ∀ᵐ (x : α) ∂μ, f x < ⊤

full_name: MeasureTheory.setLIntegral_lt_top_of_le_nnreal
code: /-- Lebesgue integral of a bounded function over a set of finite measure is finite.
Note that this lemma assumes no regularity of either `f` or `s`. -/
theorem setLIntegral_lt_top_of_le_nnreal {s : Set α} (hs : μ s ≠ ∞) {f : α → ℝ≥0∞}
    (hbdd : ∃ y : ℝ≥0, ∀ x ∈ s, f x ≤ y) : ∫⁻ x in s, f x ∂μ < ∞ 
goal: ∫⁻ (x : α) in s, f x ∂μ < ⊤

full_name: MeasureTheory.setLIntegral_lt_top_of_bddAbove
code: /-- Lebesgue integral of a bounded function over a set of finite measure is finite.
Note that this lemma assumes no regularity of either `f` or `s`. -/
theorem setLIntegral_lt_top_of_bddAbove {s : Set α} (hs : μ s ≠ ∞) {f : α → ℝ≥0}
    (hbdd : BddAbove (f '' s)) : ∫⁻ x in s, f x ∂μ < ∞ 
goal: ∫⁻ (x : α) in s, ↑(f x) ∂μ < ⊤

full_name: MeasureTheory.setLIntegral_lt_top_of_isCompact
code: theorem setLIntegral_lt_top_of_isCompact [TopologicalSpace α] {s : Set α}
    (hs : μ s ≠ ∞) (hsc : IsCompact s) {f : α → ℝ≥0} (hf : Continuous f) :
    ∫⁻ x in s, f x ∂μ < ∞ 
goal: ∫⁻ (x : α) in s, ↑(f x) ∂μ < ⊤

full_name: IsFiniteMeasure.lintegral_lt_top_of_bounded_to_ennreal
code: theorem _root_.IsFiniteMeasure.lintegral_lt_top_of_bounded_to_ennreal {α : Type*}
    [MeasurableSpace α] (μ : Measure α) [μ_fin : IsFiniteMeasure μ] {f : α → ℝ≥0∞}
    (f_bdd : ∃ c : ℝ≥0, ∀ x, f x ≤ c) : ∫⁻ x, f x ∂μ < ∞ 
goal: ∫⁻ (x : α), f x ∂μ < ⊤

full_name: MeasureTheory.tendsto_of_lintegral_tendsto_of_monotone_aux
code: /-- If a monotone sequence of functions has an upper bound and the sequence of integrals of these
functions tends to the integral of the upper bound, then the sequence of functions converges
almost everywhere to the upper bound. Auxiliary version assuming moreover that the
functions in the sequence are ae measurable. -/
lemma tendsto_of_lintegral_tendsto_of_monotone_aux {α : Type*} {mα : MeasurableSpace α}
    {f : ℕ → α → ℝ≥0∞} {F : α → ℝ≥0∞} {μ : Measure α}
    (hf_meas : ∀ n, AEMeasurable (f n) μ) (hF_meas : AEMeasurable F μ)
    (hf_tendsto : Tendsto (fun i ↦ ∫⁻ a, f i a ∂μ) atTop (𝓝 (∫⁻ a, F a ∂μ)))
    (hf_mono : ∀ᵐ a ∂μ, Monotone (fun i ↦ f i a))
    (h_bound : ∀ᵐ a ∂μ, ∀ i, f i a ≤ F a) (h_int_finite : ∫⁻ a, F a ∂μ ≠ ∞) :
    ∀ᵐ a ∂μ, Tendsto (fun i ↦ f i a) atTop (𝓝 (F a)) 
goal: ∀ᵐ (a : α) ∂μ, Filter.Tendsto (fun i => f i a) Filter.atTop (𝓝 (F a))

full_name: MeasureTheory.tendsto_of_lintegral_tendsto_of_monotone
code: /-- If a monotone sequence of functions has an upper bound and the sequence of integrals of these
functions tends to the integral of the upper bound, then the sequence of functions converges
almost everywhere to the upper bound. -/
lemma tendsto_of_lintegral_tendsto_of_monotone {α : Type*} {mα : MeasurableSpace α}
    {f : ℕ → α → ℝ≥0∞} {F : α → ℝ≥0∞} {μ : Measure α}
    (hF_meas : AEMeasurable F μ)
    (hf_tendsto : Tendsto (fun i ↦ ∫⁻ a, f i a ∂μ) atTop (𝓝 (∫⁻ a, F a ∂μ)))
    (hf_mono : ∀ᵐ a ∂μ, Monotone (fun i ↦ f i a))
    (h_bound : ∀ᵐ a ∂μ, ∀ i, f i a ≤ F a) (h_int_finite : ∫⁻ a, F a ∂μ ≠ ∞) :
    ∀ᵐ a ∂μ, Tendsto (fun i ↦ f i a) atTop (𝓝 (F a)) 
goal: ∀ᵐ (a : α) ∂μ, Filter.Tendsto (fun i => f i a) Filter.atTop (𝓝 (F a))

full_name: MeasureTheory.tendsto_of_lintegral_tendsto_of_antitone
code: /-- If an antitone sequence of functions has a lower bound and the sequence of integrals of these
functions tends to the integral of the lower bound, then the sequence of functions converges
almost everywhere to the lower bound. -/
lemma tendsto_of_lintegral_tendsto_of_antitone {α : Type*} {mα : MeasurableSpace α}
    {f : ℕ → α → ℝ≥0∞} {F : α → ℝ≥0∞} {μ : Measure α}
    (hf_meas : ∀ n, AEMeasurable (f n) μ)
    (hf_tendsto : Tendsto (fun i ↦ ∫⁻ a, f i a ∂μ) atTop (𝓝 (∫⁻ a, F a ∂μ)))
    (hf_mono : ∀ᵐ a ∂μ, Antitone (fun i ↦ f i a))
    (h_bound : ∀ᵐ a ∂μ, ∀ i, F a ≤ f i a) (h0 : ∫⁻ a, f 0 a ∂μ ≠ ∞) :
    ∀ᵐ a ∂μ, Tendsto (fun i ↦ f i a) atTop (𝓝 (F a)) 
goal: ∀ᵐ (a : α) ∂μ, Filter.Tendsto (fun i => f i a) Filter.atTop (𝓝 (F a))

full_name: MeasureTheory.exists_measurable_le_forall_setLIntegral_eq
code: /-- If `μ` is an s-finite measure, then for any function `f`
there exists a measurable function `g ≤ f`
that has the same Lebesgue integral over every set.

For the integral over the whole space, the statement is true without extra assumptions,
see `exists_measurable_le_lintegral_eq`.
See also `MeasureTheory.Measure.restrict_toMeasurable_of_sFinite` for a similar result. -/
theorem exists_measurable_le_forall_setLIntegral_eq [SFinite μ] (f : α → ℝ≥0∞) :
    ∃ g : α → ℝ≥0∞, Measurable g ∧ g ≤ f ∧ ∀ s, ∫⁻ a in s, f a ∂μ = ∫⁻ a in s, g a ∂μ 
goal: ∃ g, Measurable g ∧ g ≤ f ∧ ∀ (s : Set α), ∫⁻ (a : α) in s, f a ∂μ = ∫⁻ (a : α) in s, g a ∂μ

full_name: MeasureTheory.exists_pos_lintegral_lt_of_sigmaFinite
code: /-- In a sigma-finite measure space, there exists an integrable function which is
positive everywhere (and with an arbitrarily small integral). -/
theorem exists_pos_lintegral_lt_of_sigmaFinite (μ : Measure α) [SigmaFinite μ] {ε : ℝ≥0∞}
    (ε0 : ε ≠ 0) : ∃ g : α → ℝ≥0, (∀ x, 0 < g x) ∧ Measurable g ∧ ∫⁻ x, g x ∂μ < ε 
goal: ∃ g, (∀ (x : α), 0 < g x) ∧ Measurable g ∧ ∫⁻ (x : α), ↑(g x) ∂μ < ε

full_name: MeasureTheory.lintegral_trim
code: theorem lintegral_trim {μ : Measure α} (hm : m ≤ m0) {f : α → ℝ≥0∞} (hf : Measurable[m] f) :
    ∫⁻ a, f a ∂μ.trim hm = ∫⁻ a, f a ∂μ 
goal: ∫⁻ (a : α), f a ∂μ.trim hm = ∫⁻ (a : α), f a ∂μ

full_name: MeasureTheory.lintegral_trim_ae
code: theorem lintegral_trim_ae {μ : Measure α} (hm : m ≤ m0) {f : α → ℝ≥0∞}
    (hf : AEMeasurable f (μ.trim hm)) : ∫⁻ a, f a ∂μ.trim hm = ∫⁻ a, f a ∂μ 
goal: ∫⁻ (a : α), f a ∂μ.trim hm = ∫⁻ (a : α), f a ∂μ

full_name: MeasureTheory.univ_le_of_forall_fin_meas_le
code: theorem univ_le_of_forall_fin_meas_le {μ : Measure α} (hm : m ≤ m0) [SigmaFinite (μ.trim hm)]
    (C : ℝ≥0∞) {f : Set α → ℝ≥0∞} (hf : ∀ s, MeasurableSet[m] s → μ s ≠ ∞ → f s ≤ C)
    (h_F_lim :
      ∀ S : ℕ → Set α, (∀ n, MeasurableSet[m] (S n)) → Monotone S → f (⋃ n, S n) ≤ ⨆ n, f (S n)) :
    f univ ≤ C 
goal: f Set.univ ≤ C

full_name: MeasureTheory.lintegral_le_of_forall_fin_meas_trim_le
code: /-- If the Lebesgue integral of a function is bounded by some constant on all sets with finite
measure in a sub-σ-algebra and the measure is σ-finite on that sub-σ-algebra, then the integral
over the whole space is bounded by that same constant. -/
theorem lintegral_le_of_forall_fin_meas_trim_le {μ : Measure α} (hm : m ≤ m0)
    [SigmaFinite (μ.trim hm)] (C : ℝ≥0∞) {f : α → ℝ≥0∞}
    (hf : ∀ s, MeasurableSet[m] s → μ s ≠ ∞ → ∫⁻ x in s, f x ∂μ ≤ C) : ∫⁻ x, f x ∂μ ≤ C 
goal: ∫⁻ (x : α), f x ∂μ ≤ C

full_name: MeasureTheory.lintegral_le_of_forall_fin_meas_le
code: /-- If the Lebesgue integral of a function is bounded by some constant on all sets with finite
measure and the measure is σ-finite, then the integral over the whole space is bounded by that same
constant. -/
theorem lintegral_le_of_forall_fin_meas_le [MeasurableSpace α] {μ : Measure α} [SigmaFinite μ]
    (C : ℝ≥0∞) {f : α → ℝ≥0∞}
    (hf : ∀ s, MeasurableSet s → μ s ≠ ∞ → ∫⁻ x in s, f x ∂μ ≤ C) : ∫⁻ x, f x ∂μ ≤ C 
goal: ∫⁻ (x : α), f x ∂μ ≤ C

full_name: MeasureTheory.SimpleFunc.exists_lt_lintegral_simpleFunc_of_lt_lintegral
code: theorem SimpleFunc.exists_lt_lintegral_simpleFunc_of_lt_lintegral {m : MeasurableSpace α}
    {μ : Measure α} [SigmaFinite μ] {f : α →ₛ ℝ≥0} {L : ℝ≥0∞} (hL : L < ∫⁻ x, f x ∂μ) :
    ∃ g : α →ₛ ℝ≥0, (∀ x, g x ≤ f x) ∧ ∫⁻ x, g x ∂μ < ∞ ∧ L < ∫⁻ x, g x ∂μ 
goal: ∃ g, (∀ (x : α), ↑g x ≤ ↑f x) ∧ ∫⁻ (x : α), ↑(↑g x) ∂μ < ⊤ ∧ L < ∫⁻ (x : α), ↑(↑g x) ∂μ

full_name: MeasureTheory.exists_lt_lintegral_simpleFunc_of_lt_lintegral
code: theorem exists_lt_lintegral_simpleFunc_of_lt_lintegral {m : MeasurableSpace α} {μ : Measure α}
    [SigmaFinite μ] {f : α → ℝ≥0} {L : ℝ≥0∞} (hL : L < ∫⁻ x, f x ∂μ) :
    ∃ g : α →ₛ ℝ≥0, (∀ x, g x ≤ f x) ∧ ∫⁻ x, g x ∂μ < ∞ ∧ L < ∫⁻ x, g x ∂μ 
goal: ∃ g, (∀ (x : α), ↑g x ≤ f x) ∧ ∫⁻ (x : α), ↑(↑g x) ∂μ < ⊤ ∧ L < ∫⁻ (x : α), ↑(↑g x) ∂μ

full_name: MeasureTheory.tendsto_measure_of_ae_tendsto_indicator
code: /-- If the indicators of measurable sets `Aᵢ` tend pointwise almost everywhere to the indicator
of a measurable set `A` and we eventually have `Aᵢ ⊆ B` for some set `B` of finite measure, then
the measures of `Aᵢ` tend to the measure of `A`. -/
lemma tendsto_measure_of_ae_tendsto_indicator {μ : Measure α} (A_mble : MeasurableSet A)
    (As_mble : ∀ i, MeasurableSet (As i)) {B : Set α} (B_mble : MeasurableSet B)
    (B_finmeas : μ B ≠ ∞) (As_le_B : ∀ᶠ i in L, As i ⊆ B)
    (h_lim : ∀ᵐ x ∂μ, ∀ᶠ i in L, x ∈ As i ↔ x ∈ A) :
    Tendsto (fun i ↦ μ (As i)) L (𝓝 (μ A)) 
goal: Filter.Tendsto (fun i => μ (As i)) L (𝓝 (μ A))

full_name: MeasureTheory.tendsto_measure_of_ae_tendsto_indicator_of_isFiniteMeasure
code: /-- If `μ` is a finite measure and the indicators of measurable sets `Aᵢ` tend pointwise
almost everywhere to the indicator of a measurable set `A`, then the measures `μ Aᵢ` tend to
the measure `μ A`. -/
lemma tendsto_measure_of_ae_tendsto_indicator_of_isFiniteMeasure [IsCountablyGenerated L]
    {μ : Measure α} [IsFiniteMeasure μ] (A_mble : MeasurableSet A)
    (As_mble : ∀ i, MeasurableSet (As i)) (h_lim : ∀ᵐ x ∂μ, ∀ᶠ i in L, x ∈ As i ↔ x ∈ A) :
    Tendsto (fun i ↦ μ (As i)) L (𝓝 (μ A)) 
goal: Filter.Tendsto (fun i => μ (As i)) L (𝓝 (μ A))