mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.MeasureTheory.VectorMeasure.Decomposition.Hahn

Goal state before:
pp: α : Type u_1
inst✝ : MeasurableSpace α
s : SignedMeasure α
h : ∀ (x : ℝ), ∃ x_1 ∈ s.measureOfNegatives, x_1 < x
f : ℕ → ℝ
hf : ∀ (n : ℕ), f n ∈ s.measureOfNegatives ∧ f n < -↑n
B : ℕ → Set α
hmeas : ∀ (n : ℕ), MeasurableSet (B n)
hr : ∀ (n : ℕ), s ≤[B n] 0
h_lt : ∀ (n : ℕ), s (B n) < -↑n
A : Set α := ⋃ n, B n
hA : A = ⋃ n, B n
n : ℕ
this : s ≤[A] 0
⊢ MeasurableSet (⋃ i, B i)

Tactic:
exact MeasurableSet.iUnion hmeas

Goal state after:
Module: Mathlib.MeasureTheory.VectorMeasure.Decomposition.Jordan

Goal state before:
pp: case mp.refine_1
α : Type u_1
inst✝ : MeasurableSpace α
s : SignedMeasure α
μ : VectorMeasure α ℝ≥0∞
u : Set α
hmeas : MeasurableSet u
hu₁ : ∀ t ⊆ u, s t = 0
hu₂ : ∀ t ⊆ uᶜ, μ t = 0
i : Set α
hi₁ : MeasurableSet i
hi₂ : 0 ≤[i] s
hi₃ : s ≤[iᶜ] 0
hpos : s.toJordanDecomposition.posPart = s.toMeasureOfZeroLE i hi₁ hi₂
hneg : s.toJordanDecomposition.negPart = s.toMeasureOfLEZero iᶜ ⋯ hi₃
⊢ ↑(NNReal.mk (s (i ∩ u)) ⋯) + (s.toMeasureOfLEZero iᶜ ⋯ hi₃) u = 0

Tactic:
toMeasureOfLEZero_apply _ _ _ hmeas

Goal state after:
case mp.refine_1
α : Type u_1
inst✝ : MeasurableSpace α
s : SignedMeasure α
μ : VectorMeasure α ℝ≥0∞
u : Set α
hmeas : MeasurableSet u
hu₁ : ∀ t ⊆ u, s t = 0
hu₂ : ∀ t ⊆ uᶜ, μ t = 0
i : Set α
hi₁ : MeasurableSet i
hi₂ : 0 ≤[i] s
hi₃ : s ≤[iᶜ] 0
hpos : s.toJordanDecomposition.posPart = s.toMeasureOfZeroLE i hi₁ hi₂
hneg : s.toJordanDecomposition.negPart = s.toMeasureOfLEZero iᶜ ⋯ hi₃
⊢ ↑(NNReal.mk (s (i ∩ u)) ⋯) + ↑(NNReal.mk (-s (iᶜ ∩ u)) ⋯) = 0
Module: Mathlib.MeasureTheory.VectorMeasure.Decomposition.Lebesgue

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
μ : Measure α
t : SignedMeasure α
f : α → ℝ
hf : Measurable f
htμ : t ⟂ᵥ μ.toENNRealVectorMeasure
⊢ (t.toJordanDecomposition.posPart + μ.withDensity fun x ↦ ENNReal.ofReal (f x)) ⟂ₘ
    t.toJordanDecomposition.negPart + μ.withDensity fun x ↦ ENNReal.ofReal (-f x)

Tactic:
rw [mutuallySingular_ennreal_iff, totalVariation_mutuallySingular_iff,
    VectorMeasure.ennrealToMeasure_toENNRealVectorMeasure] at htμ

Goal state after:
α : Type u_1
m : MeasurableSpace α
μ : Measure α
t : SignedMeasure α
f : α → ℝ
hf : Measurable f
htμ : t.toJordanDecomposition.posPart ⟂ₘ μ ∧ t.toJordanDecomposition.negPart ⟂ₘ μ
⊢ (t.toJordanDecomposition.posPart + μ.withDensity fun x ↦ ENNReal.ofReal (f x)) ⟂ₘ
    t.toJordanDecomposition.negPart + μ.withDensity fun x ↦ ENNReal.ofReal (-f x)
Module: Mathlib.MeasureTheory.Function.UniformIntegrable

Goal state before:
pp: α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ : Measure α
inst✝¹ : NormedAddCommGroup β
p : ℝ≥0∞
inst✝ : IsFiniteMeasure μ
hp : 1 ≤ p
hp' : p ≠ ∞
f : ℕ → α → β
g : α → β
hf : ∀ (n : ℕ), AEStronglyMeasurable (f n) μ
hg : MemLp g p μ
hui : UnifIntegrable f p μ
hfg : ∀ᵐ (x : α) ∂μ, Tendsto (fun n ↦ f n x) atTop (𝓝 (g x))
this : ∀ (n : ℕ), eLpNorm (f n - g) p μ = eLpNorm (AEStronglyMeasurable.mk (f n) ⋯ - AEStronglyMeasurable.mk g ⋯) p μ
h_ae_forall_eq : ∀ᵐ (x : α) ∂μ, ∀ (n : ℕ), f n x = AEStronglyMeasurable.mk (f n) ⋯ x
⊢ ∀ᵐ (x : α) ∂μ, Tendsto (fun n ↦ AEStronglyMeasurable.mk (f n) ⋯ x) atTop (𝓝 (AEStronglyMeasurable.mk g ⋯ x))

Tactic:
filter_upwards [hfg, h_ae_forall_eq, hg.1.ae_eq_mk] with x hx_tendsto hxf_eq hxg_eq

Goal state after:
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ : Measure α
inst✝¹ : NormedAddCommGroup β
p : ℝ≥0∞
inst✝ : IsFiniteMeasure μ
hp : 1 ≤ p
hp' : p ≠ ∞
f : ℕ → α → β
g : α → β
hf : ∀ (n : ℕ), AEStronglyMeasurable (f n) μ
hg : MemLp g p μ
hui : UnifIntegrable f p μ
hfg : ∀ᵐ (x : α) ∂μ, Tendsto (fun n ↦ f n x) atTop (𝓝 (g x))
this : ∀ (n : ℕ), eLpNorm (f n - g) p μ = eLpNorm (AEStronglyMeasurable.mk (f n) ⋯ - AEStronglyMeasurable.mk g ⋯) p μ
h_ae_forall_eq : ∀ᵐ (x : α) ∂μ, ∀ (n : ℕ), f n x = AEStronglyMeasurable.mk (f n) ⋯ x
x : α
hx_tendsto : Tendsto (fun n ↦ f n x) atTop (𝓝 (g x))
hxf_eq : ∀ (n : ℕ), f n x = AEStronglyMeasurable.mk (f n) ⋯ x
hxg_eq : g x = AEStronglyMeasurable.mk g ⋯ x
⊢ Tendsto (fun n ↦ AEStronglyMeasurable.mk (f n) ⋯ x) atTop (𝓝 (AEStronglyMeasurable.mk g ⋯ x))
Module: Mathlib.MeasureTheory.Function.UniformIntegrable

Goal state before:
pp: α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
f : ℕ → α → β
hp : 1 ≤ p
hp' : p ≠ ∞
hf : ∀ (n : ℕ), MemLp (f n) p μ
hf_tendsto : Tendsto (fun n ↦ eLpNorm (f n) p μ) atTop (𝓝 0)
⊢ UnifIntegrable f p μ

Tactic:
intro ε

Goal state after:
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
f : ℕ → α → β
hp : 1 ≤ p
hp' : p ≠ ∞
hf : ∀ (n : ℕ), MemLp (f n) p μ
hf_tendsto : Tendsto (fun n ↦ eLpNorm (f n) p μ) atTop (𝓝 0)
ε : ℝ
⊢ 0 < ε →
    ∃ δ,
      ∃ (_ : 0 < δ),
        ∀ (i : ℕ) (s : Set α),
          MeasurableSet s → μ s ≤ ENNReal.ofReal δ → eLpNorm (s.indicator (f i)) p μ ≤ ENNReal.ofReal ε
Module: Mathlib.MeasureTheory.VectorMeasure.Decomposition.Lebesgue

Goal state before:
pp: case neg
α : Type u_1
m : MeasurableSpace α
s t : SignedMeasure α
μ : Measure α
f : α → ℝ
hf : Measurable f
htμ : t ⟂ᵥ μ.toENNRealVectorMeasure
hadd : s = t + μ.withDensityᵥ f
hfi : ¬Integrable f μ
⊢ s.HaveLebesgueDecomposition μ

Tactic:
· rw [withDensityᵥ, dif_neg hfi, add_zero] at hadd
    refine haveLebesgueDecomposition_mk' μ measurable_zero (integrable_zero _ _ μ) htμ ?_
    rwa [withDensityᵥ_zero, add_zero]

Goal state after:
Module: Mathlib.MeasureTheory.VectorMeasure.Decomposition.Lebesgue

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
μ : Measure α
s t : SignedMeasure α
f : α → ℝ
hf : Measurable f
hfi : Integrable f μ
htμ : t ⟂ᵥ μ.toENNRealVectorMeasure
hadd : s = t + μ.withDensityᵥ f
htμ' : t ⟂ᵥ μ.toENNRealVectorMeasure
⊢ t = s.singularPart μ

Tactic:
rw [mutuallySingular_ennreal_iff, totalVariation_mutuallySingular_iff,
    VectorMeasure.ennrealToMeasure_toENNRealVectorMeasure] at htμ

Goal state after:
α : Type u_1
m : MeasurableSpace α
μ : Measure α
s t : SignedMeasure α
f : α → ℝ
hf : Measurable f
hfi : Integrable f μ
htμ : t.toJordanDecomposition.posPart ⟂ₘ μ ∧ t.toJordanDecomposition.negPart ⟂ₘ μ
hadd : s = t + μ.withDensityᵥ f
htμ' : t ⟂ᵥ μ.toENNRealVectorMeasure
⊢ t = s.singularPart μ
Module: Mathlib.MeasureTheory.VectorMeasure.Decomposition.Lebesgue

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
s : SignedMeasure α
μ : Measure α
r : ℝ≥0
⊢ ((r • s).toJordanDecomposition.posPart.singularPart μ).toSignedMeasure -
      ((r • s).toJordanDecomposition.negPart.singularPart μ).toSignedMeasure =
    r • (s.toJordanDecomposition.posPart.singularPart μ).toSignedMeasure -
      r • (s.toJordanDecomposition.negPart.singularPart μ).toSignedMeasure

Tactic:
← toSignedMeasure_smul,

Goal state after:
α : Type u_1
m : MeasurableSpace α
s : SignedMeasure α
μ : Measure α
r : ℝ≥0
⊢ ((r • s).toJordanDecomposition.posPart.singularPart μ).toSignedMeasure -
      ((r • s).toJordanDecomposition.negPart.singularPart μ).toSignedMeasure =
    (r • s.toJordanDecomposition.posPart.singularPart μ).toSignedMeasure -
      r • (s.toJordanDecomposition.negPart.singularPart μ).toSignedMeasure
Module: Mathlib.MeasureTheory.Function.UniformIntegrable

Goal state before:
pp: case neg
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
hp : 1 ≤ p
hp' : p ≠ ∞
f : ι → α → β
hf : ∀ (i : ι), StronglyMeasurable (f i)
h : ∀ (ε : ℝ), 0 < ε → ∃ C, 0 < C ∧ ∀ (i : ι), eLpNorm ({x | C ≤ ‖f i x‖₊}.indicator (f i)) p μ ≤ ENNReal.ofReal ε
hpzero : p ≠ 0
hμ : ¬μ univ = 0
⊢ UnifIntegrable f p μ

Tactic:
intro ε

Goal state after:
case neg
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
hp : 1 ≤ p
hp' : p ≠ ∞
f : ι → α → β
hf : ∀ (i : ι), StronglyMeasurable (f i)
h : ∀ (ε : ℝ), 0 < ε → ∃ C, 0 < C ∧ ∀ (i : ι), eLpNorm ({x | C ≤ ‖f i x‖₊}.indicator (f i)) p μ ≤ ENNReal.ofReal ε
hpzero : p ≠ 0
hμ : ¬μ univ = 0
ε : ℝ
⊢ 0 < ε →
    ∃ δ,
      ∃ (_ : 0 < δ),
        ∀ (i : ι) (s : Set α),
          MeasurableSet s → μ s ≤ ENNReal.ofReal δ → eLpNorm (s.indicator (f i)) p μ ≤ ENNReal.ofReal ε
Module: Mathlib.MeasureTheory.Function.ConditionalExpectation.Real

Goal state before:
pp: α : Type u_1
m m0 : MeasurableSpace α
μ : Measure α
hm : m ≤ m0
hμm : SigmaFinite (μ.trim hm)
f : α → ℝ
hf : Integrable f μ
⊢ SignedMeasure.rnDeriv ((μ.withDensityᵥ f).trim hm) (μ.trim hm) =ᵐ[μ] μ[f | m]

Tactic:
refine ae_eq_condExp_of_forall_setIntegral_eq hm hf ?_ ?_ ?_
  · exact fun _ _ _ => (integrable_of_integrable_trim hm
      (SignedMeasure.integrable_rnDeriv ((μ.withDensityᵥ f).trim hm) (μ.trim hm))).integrableOn
  · intro s hs _
    conv_rhs => rw [← hf.withDensityᵥ_trim_eq_integral hm hs,
      ← SignedMeasure.withDensityᵥ_rnDeriv_eq ((μ.withDensityᵥ f).trim hm) (μ.trim hm)
        (hf.withDensityᵥ_trim_absolutelyContinuous hm)]
    rw [withDensityᵥ_apply
      (SignedMeasure.integrable_rnDeriv ((μ.withDensityᵥ f).trim hm) (μ.trim hm)) hs,
      ← setIntegral_trim hm _ hs]
    exact (SignedMeasure.measurable_rnDeriv _ _).stronglyMeasurable
  · exact (SignedMeasure.measurable_rnDeriv _ _).stronglyMeasurable.aestronglyMeasurable

Goal state after:
Module: Mathlib.MeasureTheory.VectorMeasure.Decomposition.Lebesgue

Goal state before:
pp: case a
α : Type u_1
m : MeasurableSpace α
μ : Measure α
c : ComplexMeasure α
inst✝ : c.HaveLebesgueDecomposition μ
i : Set α
hi : MeasurableSet i
⊢ ((c.singularPart μ) i + (μ.withDensityᵥ (c.rnDeriv μ)) i).im = { re := (re c) i, im := (im c) i }.im

Tactic:
rw [Complex.add_im, withDensityᵥ_apply (c.integrable_rnDeriv μ) hi, ← RCLike.im_eq_complex_im,
      ← integral_im (c.integrable_rnDeriv μ).integrableOn, RCLike.im_eq_complex_im,
      ← withDensityᵥ_apply _ hi]

Goal state after:
case a
α : Type u_1
m : MeasurableSpace α
μ : Measure α
c : ComplexMeasure α
inst✝ : c.HaveLebesgueDecomposition μ
i : Set α
hi : MeasurableSet i
⊢ ((c.singularPart μ) i).im + (μ.withDensityᵥ fun x ↦ (c.rnDeriv μ x).im) i = { re := (re c) i, im := (im c) i }.im

α : Type u_1
m : MeasurableSpace α
μ : Measure α
c : ComplexMeasure α
inst✝ : c.HaveLebesgueDecomposition μ
i : Set α
hi : MeasurableSet i
⊢ Integrable (fun x ↦ (c.rnDeriv μ x).im) μ
Module: Mathlib.MeasureTheory.Function.UniformIntegrable

Goal state before:
pp: α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝² : NormedAddCommGroup β
p : ℝ≥0∞
κ : Type u_4
u : Filter κ
inst✝¹ : u.NeBot
inst✝ : u.IsCountablyGenerated
fn : ι → α → β
hUI : UnifIntegrable fn p μ
hfn : ∀ (i : ι), AEStronglyMeasurable (fn i) μ
⊢ UnifIntegrable (fun f ↦ ↑f) p μ

Tactic:
intro ε

Goal state after:
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝² : NormedAddCommGroup β
p : ℝ≥0∞
κ : Type u_4
u : Filter κ
inst✝¹ : u.NeBot
inst✝ : u.IsCountablyGenerated
fn : ι → α → β
hUI : UnifIntegrable fn p μ
hfn : ∀ (i : ι), AEStronglyMeasurable (fn i) μ
ε : ℝ
⊢ 0 < ε →
    ∃ δ,
      ∃ (_ : 0 < δ),
        ∀ (i : ↑{g | ∃ ni, TendstoInMeasure μ (fn ∘ ni) u g}) (s : Set α),
          MeasurableSet s → μ s ≤ ENNReal.ofReal δ → eLpNorm (s.indicator ((fun f ↦ ↑f) i)) p μ ≤ ENNReal.ofReal ε
Module: Mathlib.MeasureTheory.Measure.ProbabilityMeasure

Goal state before:
pp: Ω : Type u_1
inst✝ : Nonempty Ω
m0 : MeasurableSpace Ω
μ : FiniteMeasure Ω
zero : ¬μ.mass = 0
⊢ IsProbabilityMeasure (μ.mass⁻¹ • ↑μ)

Tactic:
refine ⟨?_⟩

Goal state after:
Ω : Type u_1
inst✝ : Nonempty Ω
m0 : MeasurableSpace Ω
μ : FiniteMeasure Ω
zero : ¬μ.mass = 0
⊢ (μ.mass⁻¹ • ↑μ) univ = 1
Module: Mathlib.MeasureTheory.Function.UniformIntegrable

Goal state before:
pp: α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝² : NormedAddCommGroup β
p : ℝ≥0∞
κ : Type u_4
u : Filter κ
inst✝¹ : u.NeBot
inst✝ : u.IsCountablyGenerated
fn : ι → α → β
hUI : UnifIntegrable fn p μ
hfn : ∀ (i : ι), AEStronglyMeasurable (fn i) μ
⊢ UnifIntegrable (fun f ↦ ↑f) p μ

Tactic:
intro ε

Goal state after:
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝² : NormedAddCommGroup β
p : ℝ≥0∞
κ : Type u_4
u : Filter κ
inst✝¹ : u.NeBot
inst✝ : u.IsCountablyGenerated
fn : ι → α → β
hUI : UnifIntegrable fn p μ
hfn : ∀ (i : ι), AEStronglyMeasurable (fn i) μ
ε : ℝ
⊢ 0 < ε →
    ∃ δ,
      ∃ (_ : 0 < δ),
        ∀ (i : ↑{g | ∃ ni, ∀ᵐ (x : α) ∂μ, Tendsto (fun n ↦ fn (ni n) x) u (𝓝 (g x))}) (s : Set α),
          MeasurableSet s → μ s ≤ ENNReal.ofReal δ → eLpNorm (s.indicator ((fun f ↦ ↑f) i)) p μ ≤ ENNReal.ofReal ε
Module: Mathlib.MeasureTheory.Function.UniformIntegrable

Goal state before:
pp: case pos
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝¹ : NormedAddCommGroup β
p : ℝ≥0∞
f : ι → α → β
inst✝ : Finite ι
hp_one : 1 ≤ p
hp_top : p ≠ ∞
val✝ : Fintype ι
hι : Nonempty ι
h✝ : ∀ (i : ι), AEStronglyMeasurable (f i) μ
hf : ∀ (i : ι), eLpNorm (f i) p μ < ∞
C : ℝ≥0∞ := (Finset.image (fun i ↦ eLpNorm (f i) p μ) Finset.univ).max' ⋯
i : ι
y : ℝ≥0∞
hy : y ∈ Finset.image (fun i ↦ eLpNorm (f i) p μ) Finset.univ
⊢ y < ∞

Tactic:
Finset.mem_image

Goal state after:
case pos
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝¹ : NormedAddCommGroup β
p : ℝ≥0∞
f : ι → α → β
inst✝ : Finite ι
hp_one : 1 ≤ p
hp_top : p ≠ ∞
val✝ : Fintype ι
hι : Nonempty ι
h✝ : ∀ (i : ι), AEStronglyMeasurable (f i) μ
hf : ∀ (i : ι), eLpNorm (f i) p μ < ∞
C : ℝ≥0∞ := (Finset.image (fun i ↦ eLpNorm (f i) p μ) Finset.univ).max' ⋯
i : ι
y : ℝ≥0∞
hy : ∃ a ∈ Finset.univ, eLpNorm (f a) p μ = y
⊢ y < ∞
Module: Mathlib.MeasureTheory.Function.UniformIntegrable

Goal state before:
pp: case pos
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝¹ : NormedAddCommGroup β
p : ℝ≥0∞
f : ι → α → β
inst✝ : IsFiniteMeasure μ
hp : 1 ≤ p
hp' : p ≠ ∞
hf : ∀ (i : ι), StronglyMeasurable (f i)
h : ∀ (ε : ℝ), 0 < ε → ∃ C, ∀ (i : ι), eLpNorm ({x | C ≤ ‖f i x‖₊}.indicator (f i)) p μ ≤ ENNReal.ofReal ε
C : ℝ≥0
hC : ∀ (i : ι), eLpNorm ({x | C ≤ ‖f i x‖₊}.indicator (f i)) p μ ≤ ENNReal.ofReal 1
i : ι
x : α
hx : x ∈ {x | ‖f i x‖₊ < C}
⊢ ‖f i x‖ₑ ≤ ‖f i x + {x | C ≤ ‖f i x‖₊}.indicator (f i) x‖ₑ

Tactic:
rw [Set.indicator_of_notMem, add_zero]

Goal state after:
case pos.h
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝¹ : NormedAddCommGroup β
p : ℝ≥0∞
f : ι → α → β
inst✝ : IsFiniteMeasure μ
hp : 1 ≤ p
hp' : p ≠ ∞
hf : ∀ (i : ι), StronglyMeasurable (f i)
h : ∀ (ε : ℝ), 0 < ε → ∃ C, ∀ (i : ι), eLpNorm ({x | C ≤ ‖f i x‖₊}.indicator (f i)) p μ ≤ ENNReal.ofReal ε
C : ℝ≥0
hC : ∀ (i : ι), eLpNorm ({x | C ≤ ‖f i x‖₊}.indicator (f i)) p μ ≤ ENNReal.ofReal 1
i : ι
x : α
hx : x ∈ {x | ‖f i x‖₊ < C}
⊢ x ∉ {x | C ≤ ‖f i x‖₊}
Module: Mathlib.MeasureTheory.Measure.Tight

Goal state before:
pp: 𝓧 : Type u_1
m𝓧 : MeasurableSpace 𝓧
μ : Measure 𝓧
inst✝² : TopologicalSpace 𝓧
inst✝¹ : OpensMeasurableSpace 𝓧
inst✝ : IsFiniteMeasure μ
h : μ.InnerRegularWRT (fun s ↦ IsCompact s ∧ IsClosed[inst✝²] s) MeasurableSet
⊢ IsTightMeasureSet {μ}

Tactic:
rw [isTightMeasureSet_iff_exists_isCompact_measure_compl_le]

Goal state after:
𝓧 : Type u_1
m𝓧 : MeasurableSpace 𝓧
μ : Measure 𝓧
inst✝² : TopologicalSpace 𝓧
inst✝¹ : OpensMeasurableSpace 𝓧
inst✝ : IsFiniteMeasure μ
h : μ.InnerRegularWRT (fun s ↦ IsCompact s ∧ IsClosed[inst✝²] s) MeasurableSet
⊢ ∀ (ε : ENNReal), 0 < ε → ∃ K, IsCompact K ∧ ∀ μ_1 ∈ {μ}, μ_1 Kᶜ ≤ ε
Module: Mathlib.MeasureTheory.Measure.Tight

Goal state before:
pp: 𝓧 : Type u_1
m𝓧 : MeasurableSpace 𝓧
μ : Measure 𝓧
inst✝² : TopologicalSpace 𝓧
inst✝¹ : OpensMeasurableSpace 𝓧
inst✝ : IsFiniteMeasure μ
h : μ.InnerRegularWRT (fun s ↦ IsCompact s ∧ IsClosed[inst✝²] s) MeasurableSet
⊢ ∀ (ε : ENNReal), 0 < ε → ∃ K, IsCompact K ∧ ∀ μ_1 ∈ {μ}, μ_1 Kᶜ ≤ ε

Tactic:
intro ε

Goal state after:
𝓧 : Type u_1
m𝓧 : MeasurableSpace 𝓧
μ : Measure 𝓧
inst✝² : TopologicalSpace 𝓧
inst✝¹ : OpensMeasurableSpace 𝓧
inst✝ : IsFiniteMeasure μ
h : μ.InnerRegularWRT (fun s ↦ IsCompact s ∧ IsClosed[inst✝²] s) MeasurableSet
ε : ENNReal
⊢ 0 < ε → ∃ K, IsCompact K ∧ ∀ μ_1 ∈ {μ}, μ_1 Kᶜ ≤ ε
Module: Mathlib.MeasureTheory.Measure.Tight

Goal state before:
pp: 𝓧 : Type u_1
𝓨 : Type u_2
m𝓧 : MeasurableSpace 𝓧
S : Set (Measure 𝓧)
inst✝⁴ : TopologicalSpace 𝓧
inst✝³ : TopologicalSpace 𝓨
inst✝² : MeasurableSpace 𝓨
inst✝¹ : OpensMeasurableSpace 𝓨
inst✝ : T2Space 𝓨
hS : ∀ (ε : ENNReal), 0 < ε → ∃ K, IsCompact K ∧ ∀ μ ∈ S, μ Kᶜ ≤ ε
f : 𝓧 → 𝓨
hf : Continuous[inst✝⁴, inst✝³] f
⊢ ∀ (ε : ENNReal), 0 < ε → ∃ K, IsCompact K ∧ ∀ a ∈ S, (Measure.map f a) Kᶜ ≤ ε

Tactic:
intro ε

Goal state after:
𝓧 : Type u_1
𝓨 : Type u_2
m𝓧 : MeasurableSpace 𝓧
S : Set (Measure 𝓧)
inst✝⁴ : TopologicalSpace 𝓧
inst✝³ : TopologicalSpace 𝓨
inst✝² : MeasurableSpace 𝓨
inst✝¹ : OpensMeasurableSpace 𝓨
inst✝ : T2Space 𝓨
hS : ∀ (ε : ENNReal), 0 < ε → ∃ K, IsCompact K ∧ ∀ μ ∈ S, μ Kᶜ ≤ ε
f : 𝓧 → 𝓨
hf : Continuous[inst✝⁴, inst✝³] f
ε : ENNReal
⊢ 0 < ε → ∃ K, IsCompact K ∧ ∀ a ∈ S, (Measure.map f a) Kᶜ ≤ ε
Module: Mathlib.MeasureTheory.Measure.Tight

Goal state before:
pp: 𝓧 : Type u_1
𝓨 : Type u_2
m𝓧 : MeasurableSpace 𝓧
inst✝¹ : TopologicalSpace 𝓧
m𝓨 : MeasurableSpace 𝓨
inst✝ : TopologicalSpace 𝓨
μ : Set (Measure (𝓧 × 𝓨))
hμ₁ : ∀ (ε : ENNReal), 0 < ε → ∃ K, IsCompact K ∧ ∀ μ_1 ∈ Measure.fst '' μ, μ_1 Kᶜ ≤ ε
hμ₂ : ∀ (ε : ENNReal), 0 < ε → ∃ K, IsCompact K ∧ ∀ μ_1 ∈ Measure.snd '' μ, μ_1 Kᶜ ≤ ε
⊢ ∀ (ε : ENNReal), 0 < ε → ∃ K, IsCompact K ∧ ∀ μ_1 ∈ μ, μ_1 Kᶜ ≤ ε

Tactic:
intro ε

Goal state after:
𝓧 : Type u_1
𝓨 : Type u_2
m𝓧 : MeasurableSpace 𝓧
inst✝¹ : TopologicalSpace 𝓧
m𝓨 : MeasurableSpace 𝓨
inst✝ : TopologicalSpace 𝓨
μ : Set (Measure (𝓧 × 𝓨))
hμ₁ : ∀ (ε : ENNReal), 0 < ε → ∃ K, IsCompact K ∧ ∀ μ_1 ∈ Measure.fst '' μ, μ_1 Kᶜ ≤ ε
hμ₂ : ∀ (ε : ENNReal), 0 < ε → ∃ K, IsCompact K ∧ ∀ μ_1 ∈ Measure.snd '' μ, μ_1 Kᶜ ≤ ε
ε : ENNReal
⊢ 0 < ε → ∃ K, IsCompact K ∧ ∀ μ_1 ∈ μ, μ_1 Kᶜ ≤ ε
Module: Mathlib.MeasureTheory.Measure.Portmanteau

Goal state before:
pp: case inr
Ω : Type u_1
ι : Type u_2
L : Filter ι
inst✝³ : MeasurableSpace Ω
inst✝² : TopologicalSpace Ω
inst✝¹ : HasOuterApproxClosed Ω
inst✝ : OpensMeasurableSpace Ω
μ : FiniteMeasure Ω
μs : ι → FiniteMeasure Ω
μs_lim : Tendsto μs L (𝓝 μ)
F : Set Ω
F_closed : IsClosed[inst✝²] F
hne : L.NeBot
⊢ ∀ (ε : ℝ≥0), 0 < ε → ↑μ F < ∞ → limsup (fun i ↦ ↑(μs i) F) L ≤ ↑μ F + ↑ε

Tactic:
intro ε

Goal state after:
case inr
Ω : Type u_1
ι : Type u_2
L : Filter ι
inst✝³ : MeasurableSpace Ω
inst✝² : TopologicalSpace Ω
inst✝¹ : HasOuterApproxClosed Ω
inst✝ : OpensMeasurableSpace Ω
μ : FiniteMeasure Ω
μs : ι → FiniteMeasure Ω
μs_lim : Tendsto μs L (𝓝 μ)
F : Set Ω
F_closed : IsClosed[inst✝²] F
hne : L.NeBot
ε : ℝ≥0
⊢ 0 < ε → ↑μ F < ∞ → limsup (fun i ↦ ↑(μs i) F) L ≤ ↑μ F + ↑ε
Module: Mathlib.MeasureTheory.Measure.FiniteMeasure

Goal state before:
pp: Ω : Type u_1
Ω' : Type u_2
inst✝⁶ : MeasurableSpace Ω
inst✝⁵ : MeasurableSpace Ω'
inst✝⁴ : TopologicalSpace Ω
inst✝³ : TopologicalSpace Ω'
inst✝² : BorelSpace Ω
inst✝¹ : BorelSpace Ω'
inst✝ : NormalSpace Ω'
f : Ω → Ω'
hf : IsClosedEmbedding f
⊢ ContinuousOn (fun μ ↦ comap f μ) {μ | μ (range f)ᶜ = 0}

Tactic:
intro μ hμ
  simp only [ContinuousWithinAt]
  rw [tendsto_iff_forall_integral_tendsto]
  intro g
  obtain ⟨g', -, hg'⟩ : ∃ g' : Ω' →ᵇ ℝ, ‖g'‖ = ‖g‖ ∧ g' ∘ f = g :=
    exists_extension_norm_eq_of_isClosedEmbedding g hf
  have A x : g x = g' (f x) := by change (⇑g) x = (⇑g' ∘ f) x; simp only [hg']
  simp only [comap, toMeasure_mk, A, ← MeasurableEmbedding.integral_map hf.measurableEmbedding,
    MeasurableEmbedding.map_comap hf.measurableEmbedding]
  have B {ν : FiniteMeasure Ω'} (hν : ν (range f)ᶜ = 0) :
      ∫ y in range f, g' y ∂ν = ∫ y, g' y ∂ν := by
    congr
    simp only [null_iff_toMeasure_null] at hν
    exact Measure.restrict_eq_self_of_ae_mem hν
  rw [B hμ]
  have : Tendsto (fun (ν : FiniteMeasure Ω') ↦ ∫ y, g' y ∂ν) (𝓝[{μ | μ (range f)ᶜ = 0}] μ)
      (𝓝 (∫ (y : Ω'), g' y ∂μ)) := by
    rw [nhdsWithin]
    have A : Tendsto (fun (ν : FiniteMeasure Ω') ↦ ∫ y, g' y ∂ν) (𝓝 μ) (𝓝 (∫ (y : Ω'), g' y ∂μ)) :=
      tendsto_iff_forall_integral_tendsto.1 tendsto_id _
    exact Tendsto.mono_left A inf_le_left
  apply Tendsto.congr' _ this
  filter_upwards [self_mem_nhdsWithin] with ν hν using (B hν).symm

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

Goal state before:
pp: case refine_3.inr
α : Type u_1
m : MeasurableSpace α
μ : Measure α
p : ℝ≥0∞
E : Type u_4
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
hp : 1 ≤ p
f : ℕ → α → E
hf₁ : ∀ (i : ℕ), AEStronglyMeasurable (f i) μ
hf₂ : UnifIntegrable f p μ
C : ℝ≥0
hC : ∀ (i : ℕ), eLpNorm (f i) p μ ≤ ↑C
n : ℕ
hn : n ≠ 0
⊢ ∑ i ∈ Finset.range n, eLpNorm (f i) p μ ≤ ↑n * ↑C

Tactic:
simpa using Finset.sum_le_card_nsmul (.range n) _ _ fun i _ => hC i

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

Goal state before:
pp: Ω' : Type u_2
inst✝⁶ : MeasurableSpace Ω'
inst✝⁵ : TopologicalSpace Ω'
inst✝⁴ : BorelSpace Ω'
Ω : Type u_3
inst✝³ : MeasurableSpace Ω
inst✝² : TopologicalSpace Ω
inst✝¹ : BorelSpace Ω
inst✝ : NormalSpace Ω'
f : Ω → Ω'
hf : IsClosedEmbedding f
M : Set (FiniteMeasure Ω') := {μ | μ (range f)ᶜ = 0}
A : IsEmbedding Subtype.val
μ : FiniteMeasure Ω
s : Set Ω
hs : MeasurableSet s
⊢ (Measure.comap f (Measure.map f ↑μ)) s = ↑μ s

Tactic:
rw [Measure.comap_apply, Measure.map_apply, preimage_image_eq]

Goal state after:
case h
Ω' : Type u_2
inst✝⁶ : MeasurableSpace Ω'
inst✝⁵ : TopologicalSpace Ω'
inst✝⁴ : BorelSpace Ω'
Ω : Type u_3
inst✝³ : MeasurableSpace Ω
inst✝² : TopologicalSpace Ω
inst✝¹ : BorelSpace Ω
inst✝ : NormalSpace Ω'
f : Ω → Ω'
hf : IsClosedEmbedding f
M : Set (FiniteMeasure Ω') := {μ | μ (range f)ᶜ = 0}
A : IsEmbedding Subtype.val
μ : FiniteMeasure Ω
s : Set Ω
hs : MeasurableSet s
⊢ Function.Injective f

case hf
Ω' : Type u_2
inst✝⁶ : MeasurableSpace Ω'
inst✝⁵ : TopologicalSpace Ω'
inst✝⁴ : BorelSpace Ω'
Ω : Type u_3
inst✝³ : MeasurableSpace Ω
inst✝² : TopologicalSpace Ω
inst✝¹ : BorelSpace Ω
inst✝ : NormalSpace Ω'
f : Ω → Ω'
hf : IsClosedEmbedding f
M : Set (FiniteMeasure Ω') := {μ | μ (range f)ᶜ = 0}
A : IsEmbedding Subtype.val
μ : FiniteMeasure Ω
s : Set Ω
hs : MeasurableSet s
⊢ Measurable f

case hs
Ω' : Type u_2
inst✝⁶ : MeasurableSpace Ω'
inst✝⁵ : TopologicalSpace Ω'
inst✝⁴ : BorelSpace Ω'
Ω : Type u_3
inst✝³ : MeasurableSpace Ω
inst✝² : TopologicalSpace Ω
inst✝¹ : BorelSpace Ω
inst✝ : NormalSpace Ω'
f : Ω → Ω'
hf : IsClosedEmbedding f
M : Set (FiniteMeasure Ω') := {μ | μ (range f)ᶜ = 0}
A : IsEmbedding Subtype.val
μ : FiniteMeasure Ω
s : Set Ω
hs : MeasurableSet s
⊢ MeasurableSet (f '' s)

case hfi
Ω' : Type u_2
inst✝⁶ : MeasurableSpace Ω'
inst✝⁵ : TopologicalSpace Ω'
inst✝⁴ : BorelSpace Ω'
Ω : Type u_3
inst✝³ : MeasurableSpace Ω
inst✝² : TopologicalSpace Ω
inst✝¹ : BorelSpace Ω
inst✝ : NormalSpace Ω'
f : Ω → Ω'
hf : IsClosedEmbedding f
M : Set (FiniteMeasure Ω') := {μ | μ (range f)ᶜ = 0}
A : IsEmbedding Subtype.val
μ : FiniteMeasure Ω
s : Set Ω
hs : MeasurableSet s
⊢ Function.Injective f

case hf
Ω' : Type u_2
inst✝⁶ : MeasurableSpace Ω'
inst✝⁵ : TopologicalSpace Ω'
inst✝⁴ : BorelSpace Ω'
Ω : Type u_3
inst✝³ : MeasurableSpace Ω
inst✝² : TopologicalSpace Ω
inst✝¹ : BorelSpace Ω
inst✝ : NormalSpace Ω'
f : Ω → Ω'
hf : IsClosedEmbedding f
M : Set (FiniteMeasure Ω') := {μ | μ (range f)ᶜ = 0}
A : IsEmbedding Subtype.val
μ : FiniteMeasure Ω
s : Set Ω
hs : MeasurableSet s
⊢ ∀ (s : Set Ω), MeasurableSet s → MeasurableSet (f '' s)

case hs
Ω' : Type u_2
inst✝⁶ : MeasurableSpace Ω'
inst✝⁵ : TopologicalSpace Ω'
inst✝⁴ : BorelSpace Ω'
Ω : Type u_3
inst✝³ : MeasurableSpace Ω
inst✝² : TopologicalSpace Ω
inst✝¹ : BorelSpace Ω
inst✝ : NormalSpace Ω'
f : Ω → Ω'
hf : IsClosedEmbedding f
M : Set (FiniteMeasure Ω') := {μ | μ (range f)ᶜ = 0}
A : IsEmbedding Subtype.val
μ : FiniteMeasure Ω
s : Set Ω
hs : MeasurableSet s
⊢ MeasurableSet s