mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.MeasureTheory.Measure.Restrict

Goal state before:
pp: α : Type u_2
β : Type u_3
m0 : MeasurableSpace α
inst✝ : MeasurableSpace β
μ : Measure α
s : Set α
ν : Measure β
f : α → β
hf : QuasiMeasurePreserving f μ ν
t : Set β
hmaps : MapsTo f s t
u : Set β
hum : MeasurableSet u
⊢ (ν.restrict t) u = 0 → (map f (μ.restrict s)) u = 0

Tactic:
suffices ν (u ∩ t) = 0 → μ (f ⁻¹' u ∩ s) = 0 by simpa [hum, hf.measurable, hf.measurable hum]

Goal state after:
α : Type u_2
β : Type u_3
m0 : MeasurableSpace α
inst✝ : MeasurableSpace β
μ : Measure α
s : Set α
ν : Measure β
f : α → β
hf : QuasiMeasurePreserving f μ ν
t : Set β
hmaps : MapsTo f s t
u : Set β
hum : MeasurableSet u
⊢ ν (u ∩ t) = 0 → μ (f ⁻¹' u ∩ s) = 0
Module: Mathlib.MeasureTheory.Measure.MeasureSpace

Goal state before:
pp: α : Type u_1
R : Type u_6
m0 : MeasurableSpace α
μ : Measure α
p : α → Prop
inst✝¹ : SMul R ℝ≥0∞
inst✝ : IsScalarTower R ℝ≥0∞ ℝ≥0∞
h : ∀ᵐ (x : α) ∂μ, p x
c : R
⊢ (c • μ) {a | ¬p a} = 0

Tactic:
smul_apply,

Goal state after:
α : Type u_1
R : Type u_6
m0 : MeasurableSpace α
μ : Measure α
p : α → Prop
inst✝¹ : SMul R ℝ≥0∞
inst✝ : IsScalarTower R ℝ≥0∞ ℝ≥0∞
h : ∀ᵐ (x : α) ∂μ, p x
c : R
⊢ c • μ {a | ¬p a} = 0
Module: Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated

Goal state before:
pp: α : Type u_1
inst✝¹ : MeasurableSpace α
inst✝ : CountablyGenerated α
s : Set α
hs : s ∈ countableGeneratingSet α
⊢ MeasurableSet s

Tactic:
rw [← generateFrom_countableGeneratingSet (α := α)]

Goal state after:
α : Type u_1
inst✝¹ : MeasurableSpace α
inst✝ : CountablyGenerated α
s : Set α
hs : s ∈ countableGeneratingSet α
⊢ MeasurableSet s
Module: Mathlib.MeasureTheory.Measure.Restrict

Goal state before:
pp: α : Type u_2
β : Type u_3
m0 : MeasurableSpace α
m1 : MeasurableSpace β
f : α → β
hf : MeasurableEmbedding f
μ : Measure β
t : Set β
ht : MeasurableSet t
⊢ μ (t ∩ range f) = (μ.restrict (range f)) t

Tactic:
Measure.restrict_apply ht

Goal state after:
α : Type u_2
β : Type u_3
m0 : MeasurableSpace α
m1 : MeasurableSpace β
f : α → β
hf : MeasurableEmbedding f
μ : Measure β
t : Set β
ht : MeasurableSet t
⊢ μ (t ∩ range f) = μ (t ∩ range f)
Module: Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated

Goal state before:
pp: case refine_2
α : Type u_1
m : MeasurableSpace α
b : Set (Set α)
bct : b.Countable
hbm : ∀ s ∈ b, MeasurableSet s
hb : ∀ x ∈ univ, ∀ y ∈ univ, (∀ s ∈ b, x ∈ s ↔ y ∈ s) → x = y
⊢ SeparatesPoints α

Tactic:
rw [@separatesPoints_iff]

Goal state after:
case refine_2
α : Type u_1
m : MeasurableSpace α
b : Set (Set α)
bct : b.Countable
hbm : ∀ s ∈ b, MeasurableSet s
hb : ∀ x ∈ univ, ∀ y ∈ univ, (∀ s ∈ b, x ∈ s ↔ y ∈ s) → x = y
⊢ ∀ (x y : α), (∀ (s : Set α), MeasurableSet s → (x ∈ s ↔ y ∈ s)) → x = y
Module: Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated

Goal state before:
pp: case measurable_invFun
α : Type u_1
inst✝² : MeasurableSpace α
inst✝¹ : CountablyGenerated α
inst✝ : SeparatesPoints α
n : ℕ
⊢ ⇑(Equiv.ofInjective (mapNatBool α) ⋯) ⁻¹' Subtype.val ⁻¹' {y | y n = true} = natGeneratingSequence α n

Tactic:
simp [mapNatBool]

Goal state after:
Module: Mathlib.MeasureTheory.Measure.AEMeasurable

Goal state before:
pp: ι : Type u_1
α : Type u_2
β : Type u_3
m0 : MeasurableSpace α
inst✝¹ : MeasurableSpace β
f : α → β
inst✝ : Countable ι
μ : ι → Measure α
h : ∀ (i : ι), AEMeasurable f (μ i)
a✝ : Nontrivial β
inhabited_h : Inhabited β
s : ι → Set α := fun i ↦ toMeasurable (μ i) {x | f x ≠ mk f ⋯ x}
hsμ : ∀ (i : ι), (μ i) (s i) = 0
hsm : MeasurableSet (⋂ i, s i)
⊢ AEMeasurable f (sum μ)

Tactic:
have hs : ∀ i x, x ∉ s i → f x = (h i).mk f x := by
    intro i x hx
    contrapose! hx
    exact subset_toMeasurable _ _ hx

Goal state after:
ι : Type u_1
α : Type u_2
β : Type u_3
m0 : MeasurableSpace α
inst✝¹ : MeasurableSpace β
f : α → β
inst✝ : Countable ι
μ : ι → Measure α
h : ∀ (i : ι), AEMeasurable f (μ i)
a✝ : Nontrivial β
inhabited_h : Inhabited β
s : ι → Set α := fun i ↦ toMeasurable (μ i) {x | f x ≠ mk f ⋯ x}
hsμ : ∀ (i : ι), (μ i) (s i) = 0
hsm : MeasurableSet (⋂ i, s i)
hs : ∀ (i : ι), ∀ x ∉ s i, f x = mk f ⋯ x
⊢ AEMeasurable f (sum μ)
Module: Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated

Goal state before:
pp: case refine_1.succ
α : Type u_1
t : ℕ → Set α
n : ℕ
ih : ∀ u ∈ memPartition t n, MeasurableSet u
u : Set α
hun : u ∈ memPartition t (n + 1)
⊢ MeasurableSet u

Tactic:
simp only [memPartition_succ, mem_ofPred_eq] at hun
      obtain ⟨v, hv, huv⟩ := hun
      rcases huv with rfl | rfl
      · exact (ih v hv).inter (measurableSet_generateFrom ⟨n, rfl⟩)
      · exact (ih v hv).diff (measurableSet_generateFrom ⟨n, rfl⟩)

Goal state after:
Module: Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated

Goal state before:
pp: case refine_1
α : Type u_1
t : ℕ → Set α
u : Set α
hu : u ∈ ⋃ n, memPartition t n
⊢ MeasurableSet u

Tactic:
simp only [mem_iUnion] at hu
    obtain ⟨n, hun⟩ := hu
    induction n generalizing u with
    | zero =>
      simp only [memPartition_zero, mem_singleton_iff] at hun
      rw [hun]
      exact MeasurableSet.univ
    | succ n ih =>
      simp only [memPartition_succ, mem_ofPred_eq] at hun
      obtain ⟨v, hv, huv⟩ := hun
      rcases huv with rfl | rfl
      · exact (ih v hv).inter (measurableSet_generateFrom ⟨n, rfl⟩)
      · exact (ih v hv).diff (measurableSet_generateFrom ⟨n, rfl⟩)

Goal state after:
Module: Mathlib.MeasureTheory.Measure.AEMeasurable

Goal state before:
pp: α : Type u_2
β : Type u_3
m0 : MeasurableSpace α
μ : Measure α
f : α → β
a✝ : Nontrivial β
inhabited_h : Inhabited β
S : Set (Set β)
hSc : S.Countable
hc : MeasurableSpace.CountablyGenerated β
h : NullMeasurable f μ
T : Set β → Set α
hTf : ∀ s ∈ S, T s ⊆ f ⁻¹' s
hTm : ∀ s ∈ S, MeasurableSet (T s)
hTeq : ∀ s ∈ S, T s =ᵐ[μ] f ⁻¹' s
⊢ AEMeasurable f μ

Tactic:
choose! U hUf hUm hUeq using fun s hs ↦ (h <| .basic s hs).exists_measurable_superset_ae_eq

Goal state after:
α : Type u_2
β : Type u_3
m0 : MeasurableSpace α
μ : Measure α
f : α → β
a✝ : Nontrivial β
inhabited_h : Inhabited β
S : Set (Set β)
hSc : S.Countable
hc : MeasurableSpace.CountablyGenerated β
h : NullMeasurable f μ
T : Set β → Set α
hTf : ∀ s ∈ S, T s ⊆ f ⁻¹' s
hTm : ∀ s ∈ S, MeasurableSet (T s)
hTeq : ∀ s ∈ S, T s =ᵐ[μ] f ⁻¹' s
U : Set β → Set α
hUf : ∀ s ∈ S, U s ⊇ f ⁻¹' s
hUm : ∀ s ∈ S, MeasurableSet (U s)
hUeq : ∀ s ∈ S, U s =ᵐ[μ] f ⁻¹' s
⊢ AEMeasurable f μ
Module: Mathlib.MeasureTheory.Measure.MeasureSpace

Goal state before:
pp: α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
inst✝ : Preorder α
s : Set α
hsc : s.Countable
hst : ∀ (x : α), ∃ y ∈ s, x ≤ y
hdir : DirectedOn (fun x1 x2 ↦ x1 ≤ x2) s
⊢ μ (⋃ i ∈ s, Iic i) = μ univ

Tactic:
congr
    simp only [← bex_def] at hst
    exact iUnion₂_eq_univ_iff.2 hst

Goal state after:
Module: Mathlib.Topology.Separation.GDelta

Goal state before:
pp: case h.right
X : Type u_1
inst✝¹ : TopologicalSpace X
s t : Set X
inst✝ : NormalSpace X
st_dis : Disjoint s t
t_cl : IsClosed[inst✝¹] t
T : Set (Set X)
T_open : ∀ t ∈ T, IsOpen[inst✝¹] t
T_count : T.Countable
T_int : t = ⋂₀ T
T_nonempty : T.Nonempty
g : ℕ → ↑T
g_surj : Surjective g
g' : ℕ → Set X
g'_open : ∀ (n : ℕ), IsOpen[inst✝¹] (g' n)
clt_sub_g' : ∀ (n : ℕ), t ⊆ g' n
clg'_sub_g : ∀ (n : ℕ), closure[inst✝¹] (g' n) ⊆ ↑(g n)
clg'_int : t = ⋂ i, closure[inst✝¹] (g' i)
⊢ ∀ (n : ℕ), IsOpen[inst✝¹] (closure[inst✝¹] (g' n))ᶜ ∧ Disjoint (closure[inst✝¹] (closure[inst✝¹] (g' n))ᶜ) t

Tactic:
· refine fun n ↦ ⟨isOpen_compl_iff.mpr isClosed_closure, ?_⟩
    simp only [closure_compl, disjoint_compl_left_iff_subset]
    rw [← closure_eq_iff_isClosed.mpr t_cl] at clt_sub_g'
    exact subset_closure.trans <| (clt_sub_g' n).trans <| (g'_open n).subset_interior_closure

Goal state after:
Module: Mathlib.MeasureTheory.Constructions.BorelSpace.Basic

Goal state before:
pp: α : Type u_1
s : Set (Set α)
t : TopologicalSpace α
inst✝ : SecondCountableTopology α
hs : t = TopologicalSpace.generateFrom s
u : Set α
hu : TopologicalSpace.IsOpen u
⊢ MeasurableSet u

Tactic:
induction hu with
      | basic u hu => exact GenerateMeasurable.basic u hu
      | univ => exact @MeasurableSet.univ α (generateFrom s)
      | inter s₁ s₂ _ _ hs₁ hs₂ => exact @MeasurableSet.inter α (generateFrom s) _ _ hs₁ hs₂
      | sUnion f hf ih =>
        rcases isOpen_sUnion_countable f (by rwa [hs]) with ⟨v, hv, vf, vu⟩
        rw [← vu]
        exact @MeasurableSet.sUnion α (generateFrom s) _ hv fun x xv => ih _ (vf xv)

Goal state after:
Module: Mathlib.Topology.MetricSpace.HausdorffDistance

Goal state before:
pp: α : Type u
inst✝ : PseudoMetricSpace α
s : Set α
x y : α
h : y ∈ s
⊢ (infEDist x s).toReal ≤ (edist x y).toReal

Tactic:
exact ENNReal.toReal_mono (edist_ne_top _ _) (infEDist_le_edist_of_mem h)

Goal state after:
Module: Mathlib.MeasureTheory.Constructions.BorelSpace.Order

Goal state before:
pp: case a
α : Type u_5
inst✝³ : TopologicalSpace α
inst✝² : SecondCountableTopology α
inst✝¹ : LinearOrder α
inst✝ : OrderTopology α
⊢ borel α ≤ MeasurableSpace.generateFrom {S | ∃ l u, l ≤ u ∧ Ioc l u = S}

Tactic:
rw [borel_eq_generateFrom_Ioc]

Goal state after:
case a
α : Type u_5
inst✝³ : TopologicalSpace α
inst✝² : SecondCountableTopology α
inst✝¹ : LinearOrder α
inst✝ : OrderTopology α
⊢ MeasurableSpace.generateFrom {S | ∃ l u, l < u ∧ Ioc l u = S} ≤
    MeasurableSpace.generateFrom {S | ∃ l u, l ≤ u ∧ Ioc l u = S}
Module: Mathlib.MeasureTheory.Function.SimpleFunc

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝¹ : MeasurableSpace α
inst✝ : Zero β
f : α →ₛ β
s : Set α
hs : MeasurableSet s
t : Set β
ht : 0 ∉ t
⊢ ⇑(f.restrict s) ⁻¹' t = s ∩ ⇑f ⁻¹' t

Tactic:
by
  simp [hs, indicator_preimage_of_notMem _ _ ht, inter_comm]

Goal state after:
Module: Mathlib.MeasureTheory.Measure.Typeclasses.NullSingletonClass

Goal state before:
pp: α : Type u_1
m0 : MeasurableSpace α
s : Set α
h : s.Countable
μ : Measure α
inst✝ : NullSingletonClass μ
⊢ μ (⋃ x ∈ s, {x}) = 0

Tactic:
measure_biUnion_null_iff h

Goal state after:
α : Type u_1
m0 : MeasurableSpace α
s : Set α
h : s.Countable
μ : Measure α
inst✝ : NullSingletonClass μ
⊢ ∀ i ∈ s, μ {i} = 0
Module: Mathlib.MeasureTheory.Constructions.BorelSpace.Order

Goal state before:
pp: α : Type u_1
δ : Type u_4
inst✝⁵ : TopologicalSpace α
mα : MeasurableSpace α
inst✝⁴ : BorelSpace α
mδ : MeasurableSpace δ
inst✝³ : ConditionallyCompleteLinearOrder α
inst✝² : OrderTopology α
inst✝¹ : SecondCountableTopology α
ι : Sort u_5
μ : Measure δ
inst✝ : Countable ι
f : ι → δ → α
hf : ∀ (i : ι), AEMeasurable (f i) μ
⊢ AEMeasurable (fun b ↦ ⨆ i, f i b) μ

Tactic:
refine ⟨fun b ↦ ⨆ i, (hf i).mk (f i) b, .iSup (fun i ↦ (hf i).measurable_mk), ?_⟩
  filter_upwards [ae_all_iff.2 (fun i ↦ (hf i).ae_eq_mk)] with b hb using by simp [hb]

Goal state after:
Module: Mathlib.MeasureTheory.Function.SimpleFunc

Goal state before:
pp: α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ : Measure α
inst✝ : MeasurableSpace β
f : α →ₛ ℝ≥0∞
g : β →ₛ ℝ≥0∞
ν : Measure β
H : ∀ (y : ℝ≥0∞), μ (⇑f ⁻¹' {y}) = ν (⇑g ⁻¹' {y})
⊢ f.lintegral μ = g.lintegral ν

Tactic:
simp only [lintegral, ← H]

Goal state after:
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ : Measure α
inst✝ : MeasurableSpace β
f : α →ₛ ℝ≥0∞
g : β →ₛ ℝ≥0∞
ν : Measure β
H : ∀ (y : ℝ≥0∞), μ (⇑f ⁻¹' {y}) = ν (⇑g ⁻¹' {y})
⊢ ∑ x ∈ f.range, x * μ (⇑f ⁻¹' {x}) = ∑ x ∈ g.range, x * μ (⇑f ⁻¹' {x})
Module: Mathlib.MeasureTheory.Constructions.BorelSpace.Real

Goal state before:
pp: μ : Measure ℝ
inst✝ : IsFiniteMeasureOnCompacts μ
b : ℝ
⊢ Tendsto (fun δ ↦ μ (Icc (b - δ) (b + δ))) (𝓝[>] 0) (𝓝 (μ {b}))

Tactic:
Real.singleton_eq_inter_Icc

Goal state after:
μ : Measure ℝ
inst✝ : IsFiniteMeasureOnCompacts μ
b : ℝ
⊢ Tendsto (fun δ ↦ μ (Icc (b - δ) (b + δ))) (𝓝[>] 0) (𝓝 (μ (⋂ r, ⋂ (_ : r > 0), Icc (b - r) (b + r))))
Module: Mathlib.MeasureTheory.Function.SimpleFunc

Goal state before:
pp: α : Type u_1
mα : MeasurableSpace α
motive : (α → ℝ≥0∞) → Prop
indicator : ∀ (c : ℝ≥0∞) ⦃s : Set α⦄, MeasurableSet s → motive (s.indicator fun x ↦ c)
add :
  ∀ ⦃f g : α → ℝ≥0∞⦄,
    Disjoint (support f) (support g) → Measurable f → Measurable g → motive f → motive g → motive (f + g)
iSup :
  ∀ ⦃f : ℕ → α → ℝ≥0∞⦄,
    (∀ (n : ℕ), Measurable (f n)) → Monotone f → (∀ (n : ℕ), motive (f n)) → motive fun x ↦ ⨆ n, f n x
f : α → ℝ≥0∞
hf : Measurable f
⊢ motive f

Tactic:
convert! iSup (fun n => (eapprox f n).measurable) (monotone_eapprox f) _ using 2

Goal state after:
α : Type u_1
mα : MeasurableSpace α
motive : (α → ℝ≥0∞) → Prop
indicator : ∀ (c : ℝ≥0∞) ⦃s : Set α⦄, MeasurableSet s → motive (s.indicator fun x ↦ c)
add :
  ∀ ⦃f g : α → ℝ≥0∞⦄,
    Disjoint (support f) (support g) → Measurable f → Measurable g → motive f → motive g → motive (f + g)
iSup :
  ∀ ⦃f : ℕ → α → ℝ≥0∞⦄,
    (∀ (n : ℕ), Measurable (f n)) → Monotone f → (∀ (n : ℕ), motive (f n)) → motive fun x ↦ ⨆ n, f n x
f : α → ℝ≥0∞
hf : Measurable f
x✝ : α
⊢ f x✝ = ⨆ n, (eapprox f n) x✝

α : Type u_1
mα : MeasurableSpace α
motive : (α → ℝ≥0∞) → Prop
indicator : ∀ (c : ℝ≥0∞) ⦃s : Set α⦄, MeasurableSet s → motive (s.indicator fun x ↦ c)
add :
  ∀ ⦃f g : α → ℝ≥0∞⦄,
    Disjoint (support f) (support g) → Measurable f → Measurable g → motive f → motive g → motive (f + g)
iSup :
  ∀ ⦃f : ℕ → α → ℝ≥0∞⦄,
    (∀ (n : ℕ), Measurable (f n)) → Monotone f → (∀ (n : ℕ), motive (f n)) → motive fun x ↦ ⨆ n, f n x
f : α → ℝ≥0∞
hf : Measurable f
⊢ ∀ (n : ℕ), motive ⇑(eapprox f n)
Module: Mathlib.MeasureTheory.Function.StronglyMeasurable.AEStronglyMeasurable

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝¹ : TopologicalSpace β
m₀ : MeasurableSpace α
μ : Measure α
f : α → β
inst✝ : Zero β
s : Set α
hs : NullMeasurableSet s μ
⊢ AEStronglyMeasurable (s.indicator f) μ ↔ AEStronglyMeasurable f (μ.restrict s)

Tactic:
rw [← aestronglyMeasurable_congr (indicator_ae_eq_of_ae_eq_set hs.toMeasurable_ae_eq),
    aestronglyMeasurable_indicator_iff (measurableSet_toMeasurable ..),
    restrict_congr_set hs.toMeasurable_ae_eq]

Goal state after:
Module: Mathlib.MeasureTheory.Integral.Lebesgue.Basic

Goal state before:
pp: case pos
α : Type u_1
m : MeasurableSpace α
f : α → ℝ≥0∞
μ : Measure α
φ : α →ₛ ℝ≥0∞
hφ : ⇑φ ≤ fun a ↦ f a
ψ : α →ₛ ℝ≥0 := SimpleFunc.map ENNReal.toNNReal φ
h : ⇑(SimpleFunc.map ofNNReal ψ) =ᵐ[μ] ⇑φ
this : ∀ (x : α), ↑(ψ x) ≤ f x
⊢ φ.lintegral μ ≤ ⨆ φ, ⨆ (_ : ∀ (x : α), ↑(φ x) ≤ f x), (SimpleFunc.map ofNNReal φ).lintegral μ

Tactic:
exact le_iSup₂_of_le (φ.map ENNReal.toNNReal) this (ge_of_eq <| lintegral_congr h)

Goal state after:
Module: Mathlib.MeasureTheory.Function.StronglyMeasurable.AEStronglyMeasurable

Goal state before:
pp: case refine_2
α : Type u_1
β : Type u_2
inst✝³ : TopologicalSpace β
m₀ : MeasurableSpace α
μ : Measure α
ι : Type u_5
inst✝² : PseudoMetrizableSpace β
u : Filter ι
inst✝¹ : u.NeBot
inst✝ : u.IsCountablyGenerated
f : ι → α → β
g : α → β
hf : ∀ (i : ι), AEStronglyMeasurable (f i) μ
lim : ∀ᵐ (x : α) ∂μ, Tendsto (fun n ↦ f n x) u (𝓝 (g x))
this✝¹ : MeasurableSpace β := borel β
this✝ : BorelSpace β
⊢ ∃ t, IsSeparable t ∧ ∀ᵐ (x : α) ∂μ, g x ∈ t

Tactic:
· rcases u.exists_seq_tendsto with ⟨v, hv⟩
    have : ∀ n : ℕ, ∃ t : Set β, IsSeparable t ∧ f (v n) ⁻¹' t ∈ ae μ := fun n =>
      (aestronglyMeasurable_iff_aemeasurable_separable.1 (hf (v n))).2
    choose t t_sep ht using this
    refine ⟨closure (⋃ i, t i), .closure <| .iUnion t_sep, ?_⟩
    filter_upwards [ae_all_iff.2 ht, lim] with x hx h'x
    apply mem_closure_of_tendsto (h'x.comp hv)
    filter_upwards with n using mem_iUnion_of_mem n (hx n)

Goal state after:
Module: Mathlib.MeasureTheory.Integral.Lebesgue.Basic

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
μ : Measure α
f : α → ℝ≥0∞
hf : AEMeasurable f μ
h : ∫⁻ (a : α), f a ∂μ = 0
meas_levels_0 : ∀ ε > 0, μ {x | ε ≤ f x} = 0
u : ℕ → ℝ≥0∞
bu : ∀ (n : ℕ), u n ∈ Ioo 0 1
tu : ∀ ε > 0, ∃ N, ∀ n ≥ N, u n ≤ ε
x : α
⊢ 0 < f x ↔ ∃ i, x ∈ {x | u i ≤ f x}

Tactic:
constructor <;> intro h'

Goal state after:
case mp
α : Type u_1
m : MeasurableSpace α
μ : Measure α
f : α → ℝ≥0∞
hf : AEMeasurable f μ
h : ∫⁻ (a : α), f a ∂μ = 0
meas_levels_0 : ∀ ε > 0, μ {x | ε ≤ f x} = 0
u : ℕ → ℝ≥0∞
bu : ∀ (n : ℕ), u n ∈ Ioo 0 1
tu : ∀ ε > 0, ∃ N, ∀ n ≥ N, u n ≤ ε
x : α
h' : 0 < f x
⊢ ∃ i, x ∈ {x | u i ≤ f x}

case mpr
α : Type u_1
m : MeasurableSpace α
μ : Measure α
f : α → ℝ≥0∞
hf : AEMeasurable f μ
h : ∫⁻ (a : α), f a ∂μ = 0
meas_levels_0 : ∀ ε > 0, μ {x | ε ≤ f x} = 0
u : ℕ → ℝ≥0∞
bu : ∀ (n : ℕ), u n ∈ Ioo 0 1
tu : ∀ ε > 0, ∃ N, ∀ n ≥ N, u n ≤ ε
x : α
h' : ∃ i, x ∈ {x | u i ≤ f x}
⊢ 0 < f x
Module: Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic

Goal state before:
pp: case refine_1
α : Type u_1
β : Type u_2
f : α → β
inst✝¹ : TopologicalSpace β
inst✝ : 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 this : SigmaFinite (μ.restrict t)
S : ℕ → Set α := spanningSets (μ.restrict t)
hS_meas : ∀ (n : ℕ), MeasurableSet (S n)
f_approx : ℕ → α →ₛ β := hf_meas.approx
fs : ℕ → α →ₛ β := fun n ↦ (f_approx n).restrict (S n ∩ t)
h_fs_t_compl : ∀ (n : ℕ), ∀ x ∉ t, (fs n) x = 0
n : ℕ
y : β
hy : y ∈ ↑({y ∈ (fs n).range | y ≠ 0})
⊢ μ (S n ∩ t ∩ ⇑(f_approx n) ⁻¹' {y}) < ∞

Tactic:
refine (measure_mono Set.inter_subset_left).trans_lt ?_

Goal state after:
case refine_1
α : Type u_1
β : Type u_2
f : α → β
inst✝¹ : TopologicalSpace β
inst✝ : 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 this : SigmaFinite (μ.restrict t)
S : ℕ → Set α := spanningSets (μ.restrict t)
hS_meas : ∀ (n : ℕ), MeasurableSet (S n)
f_approx : ℕ → α →ₛ β := hf_meas.approx
fs : ℕ → α →ₛ β := fun n ↦ (f_approx n).restrict (S n ∩ t)
h_fs_t_compl : ∀ (n : ℕ), ∀ x ∉ t, (fs n) x = 0
n : ℕ
y : β
hy : y ∈ ↑({y ∈ (fs n).range | y ≠ 0})
⊢ μ (S n ∩ t) < ∞
Module: Mathlib.MeasureTheory.Integral.Lebesgue.Add

Goal state before:
pp: α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ : Measure α
inst✝ : Countable β
f : β → α → ℝ≥0∞
hf : ∀ (b : β), AEMeasurable (f b) μ
h_directed : Directed (fun x1 x2 ↦ x1 ≤ x2) f
p : α → (β → ℝ≥0∞) → Prop := fun x f' ↦ Directed LE.le f'
hp : ∀ᵐ (x : α) ∂μ, p x fun i ↦ f i x
b₁ : β
⊢ ∀ (y : β), ∃ z, aeSeq hf p b₁ ≤ aeSeq hf p z ∧ aeSeq hf p y ≤ aeSeq hf p z

Tactic:
b₂

Goal state after:
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ : Measure α
inst✝ : Countable β
f : β → α → ℝ≥0∞
hf : ∀ (b : β), AEMeasurable (f b) μ
h_directed : Directed (fun x1 x2 ↦ x1 ≤ x2) f
p : α → (β → ℝ≥0∞) → Prop := fun x f' ↦ Directed LE.le f'
hp : ∀ᵐ (x : α) ∂μ, p x fun i ↦ f i x
b₁ b₂ : β
⊢ ∃ z, aeSeq hf p b₁ ≤ aeSeq hf p z ∧ aeSeq hf p b₂ ≤ aeSeq hf p z
Module: Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic

Goal state before:
pp: case pos
α : Type u_1
β : Type u_2
f : α → β
inst✝¹ : TopologicalSpace β
inst✝ : 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 this : SigmaFinite (μ.restrict t)
S : ℕ → Set α := spanningSets (μ.restrict t)
hS_meas : ∀ (n : ℕ), MeasurableSet (S n)
f_approx : ℕ → α →ₛ β := hf_meas.approx
fs : ℕ → α →ₛ β := fun n ↦ (f_approx n).restrict (S n ∩ t)
h_fs_t_compl : ∀ (n : ℕ), ∀ x ∉ t, (fs n) x = 0
x : α
hxt : x ∈ t
h : Tendsto (fun n ↦ (f_approx n) x) atTop (𝓝 (f x))
n₁ : ℕ
hn₁ : ∀ (m_1 : ℕ), n₁ ≤ m_1 → (fs m_1) x = (f_approx m_1) x
⊢ Tendsto (fun n ↦ (fs n) x) atTop (𝓝 (f x))

Tactic:
tendsto_atTop'

Goal state after:
case pos
α : Type u_1
β : Type u_2
f : α → β
inst✝¹ : TopologicalSpace β
inst✝ : 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 this : SigmaFinite (μ.restrict t)
S : ℕ → Set α := spanningSets (μ.restrict t)
hS_meas : ∀ (n : ℕ), MeasurableSet (S n)
f_approx : ℕ → α →ₛ β := hf_meas.approx
fs : ℕ → α →ₛ β := fun n ↦ (f_approx n).restrict (S n ∩ t)
h_fs_t_compl : ∀ (n : ℕ), ∀ x ∉ t, (fs n) x = 0
x : α
hxt : x ∈ t
h : ∀ s ∈ 𝓝 (f x), ∃ a, ∀ (b : ℕ), a ≤ b → (f_approx b) x ∈ s
n₁ : ℕ
hn₁ : ∀ (m_1 : ℕ), n₁ ≤ m_1 → (fs m_1) x = (f_approx m_1) x
⊢ ∀ s ∈ 𝓝 (f x), ∃ a, ∀ (b : ℕ), a ≤ b → (fs b) x ∈ s
Module: Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic

Goal state before:
pp: α : Type u_1
β : Type u_2
γ : Type u_3
m : MeasurableSpace α
inst✝¹ : TopologicalSpace β
inst✝ : TopologicalSpace γ
f : α → β
g : α → γ
hf : StronglyMeasurable f
hg : StronglyMeasurable g
⊢ StronglyMeasurable fun x ↦ (f x, g x)

Tactic:
by
  refine ⟨fun n => SimpleFunc.pair (hf.approx n) (hg.approx n), fun x => ?_⟩
  rw [nhds_prod_eq]
  exact Tendsto.prodMk (hf.tendsto_approx x) (hg.tendsto_approx x)

Goal state after:
Module: Mathlib.MeasureTheory.Integral.Lebesgue.Sub

Goal state before:
pp: α : Type u_1
inst✝ : MeasurableSpace α
μ : Measure α
f : ℕ → α → ℝ≥0∞
F : α → ℝ≥0∞
hf : ∀ (n : ℕ), AEMeasurable (f n) μ
h_anti : ∀ᵐ (x : α) ∂μ, Antitone fun n ↦ f n x
h0 : ∫⁻ (a : α), f 0 a ∂μ ≠ ∞
h_tendsto : ∀ᵐ (x : α) ∂μ, Tendsto (fun n ↦ f n x) atTop (𝓝 (F x))
this : Antitone fun n ↦ ∫⁻ (x : α), f n x ∂μ
⊢ ∫⁻ (x : α), F x ∂μ = ⨅ n, ∫⁻ (x : α), f n x ∂μ

Tactic:
← lintegral_iInf' hf h_anti h0

Goal state after:
α : Type u_1
inst✝ : MeasurableSpace α
μ : Measure α
f : ℕ → α → ℝ≥0∞
F : α → ℝ≥0∞
hf : ∀ (n : ℕ), AEMeasurable (f n) μ
h_anti : ∀ᵐ (x : α) ∂μ, Antitone fun n ↦ f n x
h0 : ∫⁻ (a : α), f 0 a ∂μ ≠ ∞
h_tendsto : ∀ᵐ (x : α) ∂μ, Tendsto (fun n ↦ f n x) atTop (𝓝 (F x))
this : Antitone fun n ↦ ∫⁻ (x : α), f n x ∂μ
⊢ ∫⁻ (x : α), F x ∂μ = ∫⁻ (a : α), ⨅ n, f n a ∂μ
Module: Mathlib.MeasureTheory.Integral.Lebesgue.Sub

Goal state before:
pp: case neg
α : Type u_1
inst✝ : MeasurableSpace α
μ : Measure α
f : α → ℝ≥0∞
hf : ∫⁻ (a : α), f a ∂μ ≠ ∞
ε : ℝ≥0∞
hε : ε ≠ 0
hf₀ : ¬∫⁻ (a : α), f a ∂μ = 0
g : α → ℝ≥0∞
hgf : g ≤ f
hg_meas : Measurable g
hgsupp : μ (support g) < ∞
hgε : ∫⁻ (a : α), f a ∂μ - ε < ∫⁻ (a : α), g a ∂μ
⊢ ∃ s, MeasurableSet s ∧ μ s < ∞ ∧ ∫⁻ (a : α) in sᶜ, f a ∂μ < ε

Tactic:
refine ⟨_, measurableSet_support hg_meas, hgsupp, ?_⟩

Goal state after:
case neg
α : Type u_1
inst✝ : MeasurableSpace α
μ : Measure α
f : α → ℝ≥0∞
hf : ∫⁻ (a : α), f a ∂μ ≠ ∞
ε : ℝ≥0∞
hε : ε ≠ 0
hf₀ : ¬∫⁻ (a : α), f a ∂μ = 0
g : α → ℝ≥0∞
hgf : g ≤ f
hg_meas : Measurable g
hgsupp : μ (support g) < ∞
hgε : ∫⁻ (a : α), f a ∂μ - ε < ∫⁻ (a : α), g a ∂μ
⊢ ∫⁻ (a : α) in (support g)ᶜ, f a ∂μ < ε
Module: Mathlib.MeasureTheory.Integral.Lebesgue.DominatedConvergence

Goal state before:
pp: α : Type u_1
inst✝¹ : MeasurableSpace α
μ : Measure α
ι : Type u_2
l : Filter ι
inst✝ : l.IsCountablyGenerated
F : ι → α → ℝ≥0∞
f bound : α → ℝ≥0∞
hF_meas : ∀ᶠ (n : ι) in l, AEMeasurable (F n) μ
h_bound : ∀ᶠ (n : ι) in l, ∀ᵐ (a : α) ∂μ, F n a ≤ bound a
h_fin : ∫⁻ (a : α), bound a ∂μ ≠ ∞
h_lim : ∀ᵐ (a : α) ∂μ, Tendsto (fun n ↦ F n a) l (𝓝 (f a))
x : ℕ → ι
xl : Tendsto x atTop l
⊢ ?m.61

Tactic:
tendsto_atTop'

Goal state after:
α : Type u_1
inst✝¹ : MeasurableSpace α
μ : Measure α
ι : Type u_2
l : Filter ι
inst✝ : l.IsCountablyGenerated
F : ι → α → ℝ≥0∞
f bound : α → ℝ≥0∞
hF_meas : ∀ᶠ (n : ι) in l, AEMeasurable (F n) μ
h_bound : ∀ᶠ (n : ι) in l, ∀ᵐ (a : α) ∂μ, F n a ≤ bound a
h_fin : ∫⁻ (a : α), bound a ∂μ ≠ ∞
h_lim : ∀ᵐ (a : α) ∂μ, Tendsto (fun n ↦ F n a) l (𝓝 (f a))
x : ℕ → ι
xl✝ : Tendsto x atTop l
xl : ∀ s ∈ l, ∃ a, ∀ (b : ℕ), a ≤ b → x b ∈ s
⊢ ?m.61
Module: Mathlib.MeasureTheory.Integral.Lebesgue.Add

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
μ : Measure α
r : ℝ≥0∞
f : α → ℝ≥0∞
s : α →ₛ ℝ≥0∞
hs : ⇑s ≤ fun a ↦ f a
x : α
⊢ r * s x ≤ r * f x

Tactic:
hs x

Goal state after:
α : Type u_1
m : MeasurableSpace α
μ : Measure α
r : ℝ≥0∞
f : α → ℝ≥0∞
s : α →ₛ ℝ≥0∞
hs : ⇑s ≤ fun a ↦ f a
x : α
⊢ r * (fun a ↦ f a) x ≤ r * f x