Theorem: ProbabilityTheory.IdentDistrib.aestronglyMeasurable_snd
File: Mathlib/Probability/IdentDistrib.lean

Proof steps:
state_before: α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
inst✝⁶ : MeasurableSpace α
inst✝⁵ : MeasurableSpace β
inst✝⁴ : MeasurableSpace γ
inst✝³ : MeasurableSpace δ
μ : Measure α
ν : Measure β
f : α → γ
g : β → γ
inst✝² : TopologicalSpace γ
inst✝¹ : MetrizableSpace γ
inst✝ : BorelSpace γ
h : IdentDistrib f g μ ν
hf : AEStronglyMeasurable f μ
⊢ AEStronglyMeasurable g ν
state_after: α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
inst✝⁶ : MeasurableSpace α
inst✝⁵ : MeasurableSpace β
inst✝⁴ : MeasurableSpace γ
inst✝³ : MeasurableSpace δ
μ : Measure α
ν : Measure β
f : α → γ
g : β → γ
inst✝² : TopologicalSpace γ
inst✝¹ : MetrizableSpace γ
inst✝ : BorelSpace γ
h : IdentDistrib f g μ ν
hf : AEStronglyMeasurable f μ
⊢ ∃ t, IsSeparable t ∧ ∀ᵐ (x : β) ∂ν, g x ∈ t
tactic: refine aestronglyMeasurable_iff_aemeasurable_separable.2 ⟨h.aemeasurable_snd, ?_⟩
state_before: α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
inst✝⁶ : MeasurableSpace α
inst✝⁵ : MeasurableSpace β
inst✝⁴ : MeasurableSpace γ
inst✝³ : MeasurableSpace δ
μ : Measure α
ν : Measure β
f : α → γ
g : β → γ
inst✝² : TopologicalSpace γ
inst✝¹ : MetrizableSpace γ
inst✝ : BorelSpace γ
h : IdentDistrib f g μ ν
hf : AEStronglyMeasurable f μ
⊢ ∃ t, IsSeparable t ∧ ∀ᵐ (x : β) ∂ν, g x ∈ t
state_after: case intro.intro
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
inst✝⁶ : MeasurableSpace α
inst✝⁵ : MeasurableSpace β
inst✝⁴ : MeasurableSpace γ
inst✝³ : MeasurableSpace δ
μ : Measure α
ν : Measure β
f : α → γ
g : β → γ
inst✝² : TopologicalSpace γ
inst✝¹ : MetrizableSpace γ
inst✝ : BorelSpace γ
h : IdentDistrib f g μ ν
hf : AEStronglyMeasurable f μ
t : Set γ
t_sep : IsSeparable t
ht : ∀ᵐ (x : α) ∂μ, f x ∈ t
⊢ ∃ t, IsSeparable t ∧ ∀ᵐ (x : β) ∂ν, g x ∈ t
tactic: rcases (aestronglyMeasurable_iff_aemeasurable_separable.1 hf).2 with ⟨t, t_sep, ht⟩
state_before: case intro.intro
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
inst✝⁶ : MeasurableSpace α
inst✝⁵ : MeasurableSpace β
inst✝⁴ : MeasurableSpace γ
inst✝³ : MeasurableSpace δ
μ : Measure α
ν : Measure β
f : α → γ
g : β → γ
inst✝² : TopologicalSpace γ
inst✝¹ : MetrizableSpace γ
inst✝ : BorelSpace γ
h : IdentDistrib f g μ ν
hf : AEStronglyMeasurable f μ
t : Set γ
t_sep : IsSeparable t
ht : ∀ᵐ (x : α) ∂μ, f x ∈ t
⊢ ∃ t, IsSeparable t ∧ ∀ᵐ (x : β) ∂ν, g x ∈ t
state_after: case intro.intro
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
inst✝⁶ : MeasurableSpace α
inst✝⁵ : MeasurableSpace β
inst✝⁴ : MeasurableSpace γ
inst✝³ : MeasurableSpace δ
μ : Measure α
ν : Measure β
f : α → γ
g : β → γ
inst✝² : TopologicalSpace γ
inst✝¹ : MetrizableSpace γ
inst✝ : BorelSpace γ
h : IdentDistrib f g μ ν
hf : AEStronglyMeasurable f μ
t : Set γ
t_sep : IsSeparable t
ht : ∀ᵐ (x : α) ∂μ, f x ∈ t
⊢ ∀ᵐ (x : β) ∂ν, g x ∈ closure t
tactic: refine ⟨closure t, t_sep.closure, ?_⟩
state_before: case intro.intro
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
inst✝⁶ : MeasurableSpace α
inst✝⁵ : MeasurableSpace β
inst✝⁴ : MeasurableSpace γ
inst✝³ : MeasurableSpace δ
μ : Measure α
ν : Measure β
f : α → γ
g : β → γ
inst✝² : TopologicalSpace γ
inst✝¹ : MetrizableSpace γ
inst✝ : BorelSpace γ
h : IdentDistrib f g μ ν
hf : AEStronglyMeasurable f μ
t : Set γ
t_sep : IsSeparable t
ht : ∀ᵐ (x : α) ∂μ, f x ∈ t
⊢ ∀ᵐ (x : β) ∂ν, g x ∈ closure t
state_after: case intro.intro
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
inst✝⁶ : MeasurableSpace α
inst✝⁵ : MeasurableSpace β
inst✝⁴ : MeasurableSpace γ
inst✝³ : MeasurableSpace δ
μ : Measure α
ν : Measure β
f : α → γ
g : β → γ
inst✝² : TopologicalSpace γ
inst✝¹ : MetrizableSpace γ
inst✝ : BorelSpace γ
h : IdentDistrib f g μ ν
hf : AEStronglyMeasurable f μ
t : Set γ
t_sep : IsSeparable t
ht : ∀ᵐ (x : α) ∂μ, f x ∈ t
⊢ ∀ᵐ (x : α) ∂μ, f x ∈ closure t
tactic: apply h.ae_mem_snd isClosed_closure.measurableSet
state_before: case intro.intro
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
inst✝⁶ : MeasurableSpace α
inst✝⁵ : MeasurableSpace β
inst✝⁴ : MeasurableSpace γ
inst✝³ : MeasurableSpace δ
μ : Measure α
ν : Measure β
f : α → γ
g : β → γ
inst✝² : TopologicalSpace γ
inst✝¹ : MetrizableSpace γ
inst✝ : BorelSpace γ
h : IdentDistrib f g μ ν
hf : AEStronglyMeasurable f μ
t : Set γ
t_sep : IsSeparable t
ht : ∀ᵐ (x : α) ∂μ, f x ∈ t
⊢ ∀ᵐ (x : α) ∂μ, f x ∈ closure t
state_after: no goals
tactic: filter_upwards [ht] with x hx using subset_closure hx