Theorem: essSup_map_measure
File: Mathlib/MeasureTheory/Function/EssSup.lean

Proof steps:
state_before: α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ ν : Measure α
inst✝⁵ : CompleteLattice β
γ : Type u_3
mγ : MeasurableSpace γ
f : α → γ
g : γ → β
inst✝⁴ : MeasurableSpace β
inst✝³ : TopologicalSpace β
inst✝² : SecondCountableTopology β
inst✝¹ : OrderClosedTopology β
inst✝ : OpensMeasurableSpace β
hg : AEMeasurable g (Measure.map f μ)
hf : AEMeasurable f μ
⊢ essSup g (Measure.map f μ) = essSup (g ∘ f) μ
state_after: α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ ν : Measure α
inst✝⁵ : CompleteLattice β
γ : Type u_3
mγ : MeasurableSpace γ
f : α → γ
g : γ → β
inst✝⁴ : MeasurableSpace β
inst✝³ : TopologicalSpace β
inst✝² : SecondCountableTopology β
inst✝¹ : OrderClosedTopology β
inst✝ : OpensMeasurableSpace β
hg : AEMeasurable g (Measure.map f μ)
hf : AEMeasurable f μ
⊢ essSup (AEMeasurable.mk g hg ∘ f) μ = essSup (g ∘ f) μ
tactic: rw [essSup_congr_ae hg.ae_eq_mk, essSup_map_measure_of_measurable hg.measurable_mk hf]
state_before: α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ ν : Measure α
inst✝⁵ : CompleteLattice β
γ : Type u_3
mγ : MeasurableSpace γ
f : α → γ
g : γ → β
inst✝⁴ : MeasurableSpace β
inst✝³ : TopologicalSpace β
inst✝² : SecondCountableTopology β
inst✝¹ : OrderClosedTopology β
inst✝ : OpensMeasurableSpace β
hg : AEMeasurable g (Measure.map f μ)
hf : AEMeasurable f μ
⊢ essSup (AEMeasurable.mk g hg ∘ f) μ = essSup (g ∘ f) μ
state_after: α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ ν : Measure α
inst✝⁵ : CompleteLattice β
γ : Type u_3
mγ : MeasurableSpace γ
f : α → γ
g : γ → β
inst✝⁴ : MeasurableSpace β
inst✝³ : TopologicalSpace β
inst✝² : SecondCountableTopology β
inst✝¹ : OrderClosedTopology β
inst✝ : OpensMeasurableSpace β
hg : AEMeasurable g (Measure.map f μ)
hf : AEMeasurable f μ
⊢ AEMeasurable.mk g hg ∘ f =ᶠ[ae μ] g ∘ f
tactic: refine essSup_congr_ae ?_
state_before: α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ ν : Measure α
inst✝⁵ : CompleteLattice β
γ : Type u_3
mγ : MeasurableSpace γ
f : α → γ
g : γ → β
inst✝⁴ : MeasurableSpace β
inst✝³ : TopologicalSpace β
inst✝² : SecondCountableTopology β
inst✝¹ : OrderClosedTopology β
inst✝ : OpensMeasurableSpace β
hg : AEMeasurable g (Measure.map f μ)
hf : AEMeasurable f μ
⊢ AEMeasurable.mk g hg ∘ f =ᶠ[ae μ] g ∘ f
state_after: α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ ν : Measure α
inst✝⁵ : CompleteLattice β
γ : Type u_3
mγ : MeasurableSpace γ
f : α → γ
g : γ → β
inst✝⁴ : MeasurableSpace β
inst✝³ : TopologicalSpace β
inst✝² : SecondCountableTopology β
inst✝¹ : OrderClosedTopology β
inst✝ : OpensMeasurableSpace β
hg : AEMeasurable g (Measure.map f μ)
hf : AEMeasurable f μ
h_eq : ∀ᵐ (x : α) ∂μ, g (f x) = AEMeasurable.mk g hg (f x)
⊢ AEMeasurable.mk g hg ∘ f =ᶠ[ae μ] g ∘ f
tactic: have h_eq := ae_of_ae_map hf hg.ae_eq_mk
state_before: α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ ν : Measure α
inst✝⁵ : CompleteLattice β
γ : Type u_3
mγ : MeasurableSpace γ
f : α → γ
g : γ → β
inst✝⁴ : MeasurableSpace β
inst✝³ : TopologicalSpace β
inst✝² : SecondCountableTopology β
inst✝¹ : OrderClosedTopology β
inst✝ : OpensMeasurableSpace β
hg : AEMeasurable g (Measure.map f μ)
hf : AEMeasurable f μ
h_eq : ∀ᵐ (x : α) ∂μ, g (f x) = AEMeasurable.mk g hg (f x)
⊢ AEMeasurable.mk g hg ∘ f =ᶠ[ae μ] g ∘ f
state_after: α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ ν : Measure α
inst✝⁵ : CompleteLattice β
γ : Type u_3
mγ : MeasurableSpace γ
f : α → γ
g : γ → β
inst✝⁴ : MeasurableSpace β
inst✝³ : TopologicalSpace β
inst✝² : SecondCountableTopology β
inst✝¹ : OrderClosedTopology β
inst✝ : OpensMeasurableSpace β
hg : AEMeasurable g (Measure.map f μ)
hf : AEMeasurable f μ
h_eq : (fun x => g (f x)) =ᶠ[ae μ] fun x => AEMeasurable.mk g hg (f x)
⊢ AEMeasurable.mk g hg ∘ f =ᶠ[ae μ] g ∘ f
tactic: rw [← EventuallyEq] at h_eq
state_before: α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ ν : Measure α
inst✝⁵ : CompleteLattice β
γ : Type u_3
mγ : MeasurableSpace γ
f : α → γ
g : γ → β
inst✝⁴ : MeasurableSpace β
inst✝³ : TopologicalSpace β
inst✝² : SecondCountableTopology β
inst✝¹ : OrderClosedTopology β
inst✝ : OpensMeasurableSpace β
hg : AEMeasurable g (Measure.map f μ)
hf : AEMeasurable f μ
h_eq : (fun x => g (f x)) =ᶠ[ae μ] fun x => AEMeasurable.mk g hg (f x)
⊢ AEMeasurable.mk g hg ∘ f =ᶠ[ae μ] g ∘ f
state_after: no goals
tactic: exact h_eq.symm