mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic

Goal state before:
pp: case neg.inl
α : Type u_1
β : Type u_5
f : α → β
inst✝¹ : SeminormedAddCommGroup β
inst✝ : NormedSpace ℝ β
m : MeasurableSpace α
c : ℝ
hf : StronglyMeasurable f
hc : 0 ≤ c
n : ℕ
x : α
h0 : ¬‖(hf.approx n) x‖ = 0
h : ‖(hf.approx n) x‖ ≤ c
⊢ ‖min 1 (c / ‖(hf.approx n) x‖)‖ * ‖(hf.approx n) x‖ ≤ c

pp: case neg.inr
α : Type u_1
β : Type u_5
f : α → β
inst✝¹ : SeminormedAddCommGroup β
inst✝ : NormedSpace ℝ β
m : MeasurableSpace α
c : ℝ
hf : StronglyMeasurable f
hc : 0 ≤ c
n : ℕ
x : α
h0 : ¬‖(hf.approx n) x‖ = 0
h : c ≤ ‖(hf.approx n) x‖
⊢ ‖min 1 (c / ‖(hf.approx n) x‖)‖ * ‖(hf.approx n) x‖ ≤ c

Tactic:
·

Goal state after:
case neg.inl
α : Type u_1
β : Type u_5
f : α → β
inst✝¹ : SeminormedAddCommGroup β
inst✝ : NormedSpace ℝ β
m : MeasurableSpace α
c : ℝ
hf : StronglyMeasurable f
hc : 0 ≤ c
n : ℕ
x : α
h0 : ¬‖(hf.approx n) x‖ = 0
h : ‖(hf.approx n) x‖ ≤ c
⊢ ‖min 1 (c / ‖(hf.approx n) x‖)‖ * ‖(hf.approx n) x‖ ≤ c
Module: Mathlib.MeasureTheory.Function.StronglyMeasurable.AEStronglyMeasurable

Goal state before:
pp: case refine_1
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
⊢ AEMeasurable g μ

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:
·

Goal state after:
case refine_1
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
⊢ AEMeasurable g μ
Module: Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic

Goal state before:
pp: case neg.inl
α : Type u_1
β : Type u_5
f : α → β
inst✝¹ : SeminormedAddCommGroup β
inst✝ : NormedSpace ℝ β
m : MeasurableSpace α
c : ℝ
hf : StronglyMeasurable f
hc : 0 ≤ c
n : ℕ
x : α
h0 : ¬‖(hf.approx n) x‖ = 0
h : ‖(hf.approx n) x‖ ≤ c
⊢ ‖1‖ * ‖(hf.approx n) x‖ ≤ c

pp: α : Type u_1
β : Type u_5
f : α → β
inst✝¹ : SeminormedAddCommGroup β
inst✝ : NormedSpace ℝ β
m : MeasurableSpace α
c : ℝ
hf : StronglyMeasurable f
hc : 0 ≤ c
n : ℕ
x : α
h0 : ¬‖(hf.approx n) x‖ = 0
h : ‖(hf.approx n) x‖ ≤ c
⊢ 1 ≤ c / ‖(hf.approx n) x‖

Tactic:
·

Goal state after:
case neg.inl
α : Type u_1
β : Type u_5
f : α → β
inst✝¹ : SeminormedAddCommGroup β
inst✝ : NormedSpace ℝ β
m : MeasurableSpace α
c : ℝ
hf : StronglyMeasurable f
hc : 0 ≤ c
n : ℕ
x : α
h0 : ¬‖(hf.approx n) x‖ = 0
h : ‖(hf.approx n) x‖ ≤ c
⊢ ‖1‖ * ‖(hf.approx n) x‖ ≤ c
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:
·

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

Goal state before:
pp: α : Type u_1
β : Type u_5
f : α → β
inst✝¹ : SeminormedAddCommGroup β
inst✝ : NormedSpace ℝ β
m : MeasurableSpace α
c : ℝ
hf : StronglyMeasurable f
hc : 0 ≤ c
n : ℕ
x : α
h0 : ¬‖(hf.approx n) x‖ = 0
h : ‖(hf.approx n) x‖ ≤ c
⊢ 1 ≤ c / ‖(hf.approx n) x‖

Tactic:
·

Goal state after:
α : Type u_1
β : Type u_5
f : α → β
inst✝¹ : SeminormedAddCommGroup β
inst✝ : NormedSpace ℝ β
m : MeasurableSpace α
c : ℝ
hf : StronglyMeasurable f
hc : 0 ≤ c
n : ℕ
x : α
h0 : ¬‖(hf.approx n) x‖ = 0
h : ‖(hf.approx n) x‖ ≤ c
⊢ 1 ≤ c / ‖(hf.approx n) x‖
Module: Mathlib.MeasureTheory.Integral.Lebesgue.Basic

Goal state before:
pp: case refine_1
α : Type u_1
m : MeasurableSpace α
μ : Measure α
f g : α → ℝ≥0∞
h : ∀ᵐ (a : α) ∂μ, f a ≤ g a
t : Set α
hts : {x | (fun a ↦ f a ≤ g a) x}ᶜ ⊆ t
ht : MeasurableSet t
ht0 : μ t = 0
this : ∀ᵐ (x : α) ∂μ, x ∉ t
s : α →ₛ ℝ≥0∞
hfs : ⇑s ≤ fun a ↦ f a
⊢ ⇑(s.restrict tᶜ) ≤ fun a ↦ g a

pp: case refine_2
α : Type u_1
m : MeasurableSpace α
μ : Measure α
f g : α → ℝ≥0∞
h : ∀ᵐ (a : α) ∂μ, f a ≤ g a
t : Set α
hts : {x | (fun a ↦ f a ≤ g a) x}ᶜ ⊆ t
ht : MeasurableSet t
ht0 : μ t = 0
this : ∀ᵐ (x : α) ∂μ, x ∉ t
s : α →ₛ ℝ≥0∞
hfs : ⇑s ≤ fun a ↦ f a
⊢ s.lintegral μ ≤ (s.restrict tᶜ).lintegral μ

Tactic:
·

Goal state after:
case refine_1
α : Type u_1
m : MeasurableSpace α
μ : Measure α
f g : α → ℝ≥0∞
h : ∀ᵐ (a : α) ∂μ, f a ≤ g a
t : Set α
hts : {x | (fun a ↦ f a ≤ g a) x}ᶜ ⊆ t
ht : MeasurableSet t
ht0 : μ t = 0
this : ∀ᵐ (x : α) ∂μ, x ∉ t
s : α →ₛ ℝ≥0∞
hfs : ⇑s ≤ fun a ↦ f a
⊢ ⇑(s.restrict tᶜ) ≤ fun a ↦ g a
Module: Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic

Goal state before:
pp: case neg.inr
α : Type u_1
β : Type u_5
f : α → β
inst✝¹ : SeminormedAddCommGroup β
inst✝ : NormedSpace ℝ β
m : MeasurableSpace α
c : ℝ
hf : StronglyMeasurable f
hc : 0 ≤ c
n : ℕ
x : α
h0 : ¬‖(hf.approx n) x‖ = 0
h : c ≤ ‖(hf.approx n) x‖
⊢ ‖min 1 (c / ‖(hf.approx n) x‖)‖ * ‖(hf.approx n) x‖ ≤ c

Tactic:
·

Goal state after:
case neg.inr
α : Type u_1
β : Type u_5
f : α → β
inst✝¹ : SeminormedAddCommGroup β
inst✝ : NormedSpace ℝ β
m : MeasurableSpace α
c : ℝ
hf : StronglyMeasurable f
hc : 0 ≤ c
n : ℕ
x : α
h0 : ¬‖(hf.approx n) x‖ = 0
h : c ≤ ‖(hf.approx n) x‖
⊢ ‖min 1 (c / ‖(hf.approx n) x‖)‖ * ‖(hf.approx n) x‖ ≤ c
Module: Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic

Goal state before:
pp: case neg.inr
α : Type u_1
β : Type u_5
f : α → β
inst✝¹ : SeminormedAddCommGroup β
inst✝ : NormedSpace ℝ β
m : MeasurableSpace α
c : ℝ
hf : StronglyMeasurable f
hc : 0 ≤ c
n : ℕ
x : α
h0 : ¬‖(hf.approx n) x‖ = 0
h : c ≤ ‖(hf.approx n) x‖
⊢ ‖c / ‖(hf.approx n) x‖‖ * ‖(hf.approx n) x‖ ≤ c

pp: α : Type u_1
β : Type u_5
f : α → β
inst✝¹ : SeminormedAddCommGroup β
inst✝ : NormedSpace ℝ β
m : MeasurableSpace α
c : ℝ
hf : StronglyMeasurable f
hc : 0 ≤ c
n : ℕ
x : α
h0 : ¬‖(hf.approx n) x‖ = 0
h : c ≤ ‖(hf.approx n) x‖
⊢ c / ‖(hf.approx n) x‖ ≤ 1

Tactic:
·

Goal state after:
case neg.inr
α : Type u_1
β : Type u_5
f : α → β
inst✝¹ : SeminormedAddCommGroup β
inst✝ : NormedSpace ℝ β
m : MeasurableSpace α
c : ℝ
hf : StronglyMeasurable f
hc : 0 ≤ c
n : ℕ
x : α
h0 : ¬‖(hf.approx n) x‖ = 0
h : c ≤ ‖(hf.approx n) x‖
⊢ ‖c / ‖(hf.approx n) x‖‖ * ‖(hf.approx n) x‖ ≤ c
Module: Mathlib.MeasureTheory.Integral.Lebesgue.Basic

Goal state before:
pp: case refine_2
α : Type u_1
m : MeasurableSpace α
μ : Measure α
f g : α → ℝ≥0∞
h : ∀ᵐ (a : α) ∂μ, f a ≤ g a
t : Set α
hts : {x | (fun a ↦ f a ≤ g a) x}ᶜ ⊆ t
ht : MeasurableSet t
ht0 : μ t = 0
this : ∀ᵐ (x : α) ∂μ, x ∉ t
s : α →ₛ ℝ≥0∞
hfs : ⇑s ≤ fun a ↦ f a
⊢ s.lintegral μ ≤ (s.restrict tᶜ).lintegral μ

Tactic:
·

Goal state after:
case refine_2
α : Type u_1
m : MeasurableSpace α
μ : Measure α
f g : α → ℝ≥0∞
h : ∀ᵐ (a : α) ∂μ, f a ≤ g a
t : Set α
hts : {x | (fun a ↦ f a ≤ g a) x}ᶜ ⊆ t
ht : MeasurableSet t
ht0 : μ t = 0
this : ∀ᵐ (x : α) ∂μ, x ∉ t
s : α →ₛ ℝ≥0∞
hfs : ⇑s ≤ fun a ↦ f a
⊢ s.lintegral μ ≤ (s.restrict tᶜ).lintegral μ
Module: Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic

Goal state before:
pp: α : Type u_1
β : Type u_5
f : α → β
inst✝¹ : SeminormedAddCommGroup β
inst✝ : NormedSpace ℝ β
m : MeasurableSpace α
c : ℝ
hf : StronglyMeasurable f
hc : 0 ≤ c
n : ℕ
x : α
h0 : ¬‖(hf.approx n) x‖ = 0
h : c ≤ ‖(hf.approx n) x‖
⊢ c / ‖(hf.approx n) x‖ ≤ 1

Tactic:
·

Goal state after:
α : Type u_1
β : Type u_5
f : α → β
inst✝¹ : SeminormedAddCommGroup β
inst✝ : NormedSpace ℝ β
m : MeasurableSpace α
c : ℝ
hf : StronglyMeasurable f
hc : 0 ≤ c
n : ℕ
x : α
h0 : ¬‖(hf.approx n) x‖ = 0
h : c ≤ ‖(hf.approx n) x‖
⊢ c / ‖(hf.approx n) x‖ ≤ 1
Module: Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic

Goal state before:
pp: case inl
α : Type u_1
β : Type u_2
f : α → β
inst✝² : TopologicalSpace β
inst✝¹ : Nonempty β
inst✝ : T2Space β
hα : IsEmpty α
⊢ StronglyMeasurable f ↔ ∃ c, f = fun x ↦ c

pp: case inr
α : Type u_1
β : Type u_2
f : α → β
inst✝² : TopologicalSpace β
inst✝¹ : Nonempty β
inst✝ : T2Space β
hα : Nonempty α
⊢ StronglyMeasurable f ↔ ∃ c, f = fun x ↦ c

Tactic:
·

Goal state after:
case inl
α : Type u_1
β : Type u_2
f : α → β
inst✝² : TopologicalSpace β
inst✝¹ : Nonempty β
inst✝ : T2Space β
hα : IsEmpty α
⊢ StronglyMeasurable f ↔ ∃ c, f = fun x ↦ c
Module: Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic

Goal state before:
pp: case inr.refine_1
α : Type u_1
β : Type u_2
f : α → β
inst✝² : TopologicalSpace β
inst✝¹ : Nonempty β
inst✝ : T2Space β
hα : Nonempty α
hf : StronglyMeasurable f
⊢ ∃ c, f = fun x ↦ c

pp: case inr.refine_2
α : Type u_1
β : Type u_2
f : α → β
inst✝² : TopologicalSpace β
inst✝¹ : Nonempty β
inst✝ : T2Space β
hα : Nonempty α
hf_eq : ∃ c, f = fun x ↦ c
⊢ StronglyMeasurable f

Tactic:
·

Goal state after:
case inr.refine_1
α : Type u_1
β : Type u_2
f : α → β
inst✝² : TopologicalSpace β
inst✝¹ : Nonempty β
inst✝ : T2Space β
hα : Nonempty α
hf : StronglyMeasurable f
⊢ ∃ c, f = fun x ↦ c
Module: Mathlib.MeasureTheory.Integral.Lebesgue.Basic

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
μ : Measure α
s : Set α
f g : α → ℝ≥0∞
hg : AEMeasurable g (μ.restrict s)
hfg : ∀ᵐ (x : α) ∂μ, x ∈ s → f x ≤ g x
f' : α → ℝ≥0∞
hf'm : Measurable f'
hle : f' ≤ f
hf' : ∫⁻ (a : α) in s, f a ∂μ = ∫⁻ (a : α) in s, f' a ∂μ
⊢ ∀ᵐ (x : α) ∂μ, x ∈ s → f' x ≤ g x

pp: α : Type u_1
m : MeasurableSpace α
μ : Measure α
s : Set α
f g : α → ℝ≥0∞
hg : AEMeasurable g (μ.restrict s)
hfg : ∀ᵐ (x : α) ∂μ, x ∈ s → f x ≤ g x
f' : α → ℝ≥0∞
hf'm : Measurable f'
hle : f' ≤ f
hf' : ∫⁻ (a : α) in s, f a ∂μ = ∫⁻ (a : α) in s, f' a ∂μ
⊢ NullMeasurableSet {x | f' x ≤ g x} (μ.restrict s)

Tactic:
·

Goal state after:
α : Type u_1
m : MeasurableSpace α
μ : Measure α
s : Set α
f g : α → ℝ≥0∞
hg : AEMeasurable g (μ.restrict s)
hfg : ∀ᵐ (x : α) ∂μ, x ∈ s → f x ≤ g x
f' : α → ℝ≥0∞
hf'm : Measurable f'
hle : f' ≤ f
hf' : ∫⁻ (a : α) in s, f a ∂μ = ∫⁻ (a : α) in s, f' a ∂μ
⊢ ∀ᵐ (x : α) ∂μ, x ∈ s → f' x ≤ g x
Module: Mathlib.MeasureTheory.Integral.Lebesgue.Basic

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
μ : Measure α
s : Set α
f g : α → ℝ≥0∞
hg : AEMeasurable g (μ.restrict s)
hfg : ∀ᵐ (x : α) ∂μ, x ∈ s → f x ≤ g x
f' : α → ℝ≥0∞
hf'm : Measurable f'
hle : f' ≤ f
hf' : ∫⁻ (a : α) in s, f a ∂μ = ∫⁻ (a : α) in s, f' a ∂μ
⊢ NullMeasurableSet {x | f' x ≤ g x} (μ.restrict s)

Tactic:
·

Goal state after:
α : Type u_1
m : MeasurableSpace α
μ : Measure α
s : Set α
f g : α → ℝ≥0∞
hg : AEMeasurable g (μ.restrict s)
hfg : ∀ᵐ (x : α) ∂μ, x ∈ s → f x ≤ g x
f' : α → ℝ≥0∞
hf'm : Measurable f'
hle : f' ≤ f
hf' : ∫⁻ (a : α) in s, f a ∂μ = ∫⁻ (a : α) in s, f' a ∂μ
⊢ NullMeasurableSet {x | f' x ≤ g x} (μ.restrict s)
Module: Mathlib.MeasureTheory.Function.StronglyMeasurable.AEStronglyMeasurable

Goal state before:
pp: case refine_1
α : Type u_1
β : Type u_2
inst✝¹ : TopologicalSpace β
m₀ : MeasurableSpace α
μ : Measure α
f g : α → β
s : Set α
inst✝ : DecidablePred fun x ↦ x ∈ s
hs : MeasurableSet s
hf : AEStronglyMeasurable f (μ.restrict s)
hg : AEStronglyMeasurable g (μ.restrict sᶜ)
⊢ ∀ᵐ (x : α) ∂μ.restrict s,
    s.piecewise f g x = s.piecewise (AEStronglyMeasurable.mk f hf) (AEStronglyMeasurable.mk g hg) x

pp: case refine_2
α : Type u_1
β : Type u_2
inst✝¹ : TopologicalSpace β
m₀ : MeasurableSpace α
μ : Measure α
f g : α → β
s : Set α
inst✝ : DecidablePred fun x ↦ x ∈ s
hs : MeasurableSet s
hf : AEStronglyMeasurable f (μ.restrict s)
hg : AEStronglyMeasurable g (μ.restrict sᶜ)
⊢ ∀ᵐ (x : α) ∂μ.restrict sᶜ,
    s.piecewise f g x = s.piecewise (AEStronglyMeasurable.mk f hf) (AEStronglyMeasurable.mk g hg) x

Tactic:
·

Goal state after:
case refine_1
α : Type u_1
β : Type u_2
inst✝¹ : TopologicalSpace β
m₀ : MeasurableSpace α
μ : Measure α
f g : α → β
s : Set α
inst✝ : DecidablePred fun x ↦ x ∈ s
hs : MeasurableSet s
hf : AEStronglyMeasurable f (μ.restrict s)
hg : AEStronglyMeasurable g (μ.restrict sᶜ)
⊢ ∀ᵐ (x : α) ∂μ.restrict s,
    s.piecewise f g x = s.piecewise (AEStronglyMeasurable.mk f hf) (AEStronglyMeasurable.mk g hg) x
Module: Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic

Goal state before:
pp: case inr.refine_2
α : Type u_1
β : Type u_2
f : α → β
inst✝² : TopologicalSpace β
inst✝¹ : Nonempty β
inst✝ : T2Space β
hα : Nonempty α
hf_eq : ∃ c, f = fun x ↦ c
⊢ StronglyMeasurable f

Tactic:
·

Goal state after:
case inr.refine_2
α : Type u_1
β : Type u_2
f : α → β
inst✝² : TopologicalSpace β
inst✝¹ : Nonempty β
inst✝ : T2Space β
hα : Nonempty α
hf_eq : ∃ c, f = fun x ↦ c
⊢ StronglyMeasurable f
Module: Mathlib.MeasureTheory.Function.StronglyMeasurable.AEStronglyMeasurable

Goal state before:
pp: case refine_2
α : Type u_1
β : Type u_2
inst✝¹ : TopologicalSpace β
m₀ : MeasurableSpace α
μ : Measure α
f g : α → β
s : Set α
inst✝ : DecidablePred fun x ↦ x ∈ s
hs : MeasurableSet s
hf : AEStronglyMeasurable f (μ.restrict s)
hg : AEStronglyMeasurable g (μ.restrict sᶜ)
⊢ ∀ᵐ (x : α) ∂μ.restrict sᶜ,
    s.piecewise f g x = s.piecewise (AEStronglyMeasurable.mk f hf) (AEStronglyMeasurable.mk g hg) x

Tactic:
·

Goal state after:
case refine_2
α : Type u_1
β : Type u_2
inst✝¹ : TopologicalSpace β
m₀ : MeasurableSpace α
μ : Measure α
f g : α → β
s : Set α
inst✝ : DecidablePred fun x ↦ x ∈ s
hs : MeasurableSet s
hf : AEStronglyMeasurable f (μ.restrict s)
hg : AEStronglyMeasurable g (μ.restrict sᶜ)
⊢ ∀ᵐ (x : α) ∂μ.restrict sᶜ,
    s.piecewise f g x = s.piecewise (AEStronglyMeasurable.mk f hf) (AEStronglyMeasurable.mk g hg) 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), μ (support ⇑(fs n)) < ∞

pp: case refine_2
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
⊢ Tendsto (fun n ↦ (fs n) x) atTop (𝓝 (f x))

Tactic:
·

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), μ (support ⇑(fs n)) < ∞
Module: Mathlib.MeasureTheory.Integral.Lebesgue.Basic

Goal state before:
pp: 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}

pp: 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

Tactic:
·

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}
Module: Mathlib.MeasureTheory.Integral.Lebesgue.Add

Goal state before:
pp: case pos
α : Type u_1
m✝ : MeasurableSpace α
μ : Measure α
f : ℕ → α → ℝ≥0∞
hf : ∀ (n : ℕ), AEMeasurable (f n) μ
h_mono : ∀ᵐ (x : α) ∂μ, Monotone fun n ↦ f n x
p : α → (ℕ → ℝ≥0∞) → Prop := fun x f' ↦ Monotone f'
hp : ∀ᵐ (x : α) ∂μ, p x fun i ↦ f i x
n m : ℕ
hnm : n ≤ m
x : α
hx : x ∈ aeSeqSet hf p
⊢ aeSeq hf p n x ≤ aeSeq hf p m x

pp: case neg
α : Type u_1
m✝ : MeasurableSpace α
μ : Measure α
f : ℕ → α → ℝ≥0∞
hf : ∀ (n : ℕ), AEMeasurable (f n) μ
h_mono : ∀ᵐ (x : α) ∂μ, Monotone fun n ↦ f n x
p : α → (ℕ → ℝ≥0∞) → Prop := fun x f' ↦ Monotone f'
hp : ∀ᵐ (x : α) ∂μ, p x fun i ↦ f i x
n m : ℕ
hnm : n ≤ m
x : α
hx : x ∉ aeSeqSet hf p
⊢ aeSeq hf p n x ≤ aeSeq hf p m x

Tactic:
·

Goal state after:
case pos
α : Type u_1
m✝ : MeasurableSpace α
μ : Measure α
f : ℕ → α → ℝ≥0∞
hf : ∀ (n : ℕ), AEMeasurable (f n) μ
h_mono : ∀ᵐ (x : α) ∂μ, Monotone fun n ↦ f n x
p : α → (ℕ → ℝ≥0∞) → Prop := fun x f' ↦ Monotone f'
hp : ∀ᵐ (x : α) ∂μ, p x fun i ↦ f i x
n m : ℕ
hnm : n ≤ m
x : α
hx : x ∈ aeSeqSet hf p
⊢ aeSeq hf p n x ≤ aeSeq hf p m x
Module: Mathlib.MeasureTheory.Integral.Lebesgue.Basic

Goal state before:
pp: 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

Tactic:
·

Goal state after:
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.Integral.Lebesgue.Add

Goal state before:
pp: case neg
α : Type u_1
m✝ : MeasurableSpace α
μ : Measure α
f : ℕ → α → ℝ≥0∞
hf : ∀ (n : ℕ), AEMeasurable (f n) μ
h_mono : ∀ᵐ (x : α) ∂μ, Monotone fun n ↦ f n x
p : α → (ℕ → ℝ≥0∞) → Prop := fun x f' ↦ Monotone f'
hp : ∀ᵐ (x : α) ∂μ, p x fun i ↦ f i x
n m : ℕ
hnm : n ≤ m
x : α
hx : x ∉ aeSeqSet hf p
⊢ aeSeq hf p n x ≤ aeSeq hf p m x

Tactic:
·

Goal state after:
case neg
α : Type u_1
m✝ : MeasurableSpace α
μ : Measure α
f : ℕ → α → ℝ≥0∞
hf : ∀ (n : ℕ), AEMeasurable (f n) μ
h_mono : ∀ᵐ (x : α) ∂μ, Monotone fun n ↦ f n x
p : α → (ℕ → ℝ≥0∞) → Prop := fun x f' ↦ Monotone f'
hp : ∀ᵐ (x : α) ∂μ, p x fun i ↦ f i x
n m : ℕ
hnm : n ≤ m
x : α
hx : x ∉ aeSeqSet hf p
⊢ aeSeq hf p n x ≤ aeSeq hf p m 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})
⊢ y ≠ 0

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:
·

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})
⊢ y ≠ 0
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✝ : PseudoMetrizableSpace β
s t : Set α
⊢ AEStronglyMeasurable f (μ.restrict (s ∪ t)) ↔
    AEStronglyMeasurable f (μ.restrict s) ∧ AEStronglyMeasurable f (μ.restrict t)

Tactic:
by
  simp only [union_eq_iUnion, aestronglyMeasurable_iUnion_iff, Bool.forall_bool, cond, and_comm]

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

Goal state before:
pp: case refine_2
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
⊢ Tendsto (fun n ↦ (fs n) x) atTop (𝓝 (f x))

Tactic:
·

Goal state after:
case refine_2
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
⊢ Tendsto (fun n ↦ (fs n) x) atTop (𝓝 (f x))