Module: Mathlib.Analysis.Convex.Integral

Goal state before:
pp: α : Type u_1
E : Type u_2
m0 : MeasurableSpace α
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : CompleteSpace E
μ : Measure α
s : Set E
f : α → E
inst✝ : IsProbabilityMeasure μ
hs : Convex ℝ s
hsc : IsClosed[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] s
hf : ∀ᵐ (x : α) ∂μ, f x ∈ s
this✝² : MeasurableSpace E := borel E
this✝¹ : BorelSpace E
g : α → E
hfi : Integrable g μ
hgm : StronglyMeasurable g
hfg : f =ᵐ[μ] g
this✝ : SeparableSpace ↑(range g ∩ s)
y₀ : E
h₀ : y₀ ∈ range g ∩ s
hg : ∀ᵐ (x : α) ∂μ, g x ∈ closure[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (range g ∩ s)
G : ℕ → SimpleFunc α E := SimpleFunc.approxOn g ⋯ (range g ∩ s) y₀ h₀
this : Tendsto (fun n ↦ SimpleFunc.integral μ (G n)) atTop (𝓝 (∫ (x : α), g x ∂μ))
⊢ ∫ (a : α), g a ∂μ ∈ s

Tactic:
refine hsc.mem_of_tendsto this (Eventually.of_forall fun n => hs.sum_mem ?_ ?_ ?_)

Goal state after:
case refine_1
α : Type u_1
E : Type u_2
m0 : MeasurableSpace α
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : CompleteSpace E
μ : Measure α
s : Set E
f : α → E
inst✝ : IsProbabilityMeasure μ
hs : Convex ℝ s
hsc : IsClosed[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] s
hf : ∀ᵐ (x : α) ∂μ, f x ∈ s
this✝² : MeasurableSpace E := borel E
this✝¹ : BorelSpace E
g : α → E
hfi : Integrable g μ
hgm : StronglyMeasurable g
hfg : f =ᵐ[μ] g
this✝ : SeparableSpace ↑(range g ∩ s)
y₀ : E
h₀ : y₀ ∈ range g ∩ s
hg : ∀ᵐ (x : α) ∂μ, g x ∈ closure[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (range g ∩ s)
G : ℕ → SimpleFunc α E := SimpleFunc.approxOn g ⋯ (range g ∩ s) y₀ h₀
this : Tendsto (fun n ↦ SimpleFunc.integral μ (G n)) atTop (𝓝 (∫ (x : α), g x ∂μ))
n : ℕ
⊢ ∀ i ∈ (G n).range, 0 ≤ μ.real (⇑(G n) ⁻¹' {i})

case refine_2
α : Type u_1
E : Type u_2
m0 : MeasurableSpace α
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : CompleteSpace E
μ : Measure α
s : Set E
f : α → E
inst✝ : IsProbabilityMeasure μ
hs : Convex ℝ s
hsc : IsClosed[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] s
hf : ∀ᵐ (x : α) ∂μ, f x ∈ s
this✝² : MeasurableSpace E := borel E
this✝¹ : BorelSpace E
g : α → E
hfi : Integrable g μ
hgm : StronglyMeasurable g
hfg : f =ᵐ[μ] g
this✝ : SeparableSpace ↑(range g ∩ s)
y₀ : E
h₀ : y₀ ∈ range g ∩ s
hg : ∀ᵐ (x : α) ∂μ, g x ∈ closure[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (range g ∩ s)
G : ℕ → SimpleFunc α E := SimpleFunc.approxOn g ⋯ (range g ∩ s) y₀ h₀
this : Tendsto (fun n ↦ SimpleFunc.integral μ (G n)) atTop (𝓝 (∫ (x : α), g x ∂μ))
n : ℕ
⊢ ∑ i ∈ (G n).range, μ.real (⇑(G n) ⁻¹' {i}) = 1

case refine_3
α : Type u_1
E : Type u_2
m0 : MeasurableSpace α
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : CompleteSpace E
μ : Measure α
s : Set E
f : α → E
inst✝ : IsProbabilityMeasure μ
hs : Convex ℝ s
hsc : IsClosed[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] s
hf : ∀ᵐ (x : α) ∂μ, f x ∈ s
this✝² : MeasurableSpace E := borel E
this✝¹ : BorelSpace E
g : α → E
hfi : Integrable g μ
hgm : StronglyMeasurable g
hfg : f =ᵐ[μ] g
this✝ : SeparableSpace ↑(range g ∩ s)
y₀ : E
h₀ : y₀ ∈ range g ∩ s
hg : ∀ᵐ (x : α) ∂μ, g x ∈ closure[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (range g ∩ s)
G : ℕ → SimpleFunc α E := SimpleFunc.approxOn g ⋯ (range g ∩ s) y₀ h₀
this : Tendsto (fun n ↦ SimpleFunc.integral μ (G n)) atTop (𝓝 (∫ (x : α), g x ∂μ))
n : ℕ
⊢ ∀ i ∈ (G n).range, ↑(ContinuousLinearMap.id ℝ E) i ∈ s