Theorem: MeasureTheory.AEStronglyMeasurable'.sub
File: Mathlib/MeasureTheory/Function/ConditionalExpectation/AEMeasurable.lean

Proof steps:
state_before: α : Type u_1
β : Type u_2
𝕜 : Type u_3
m m0 : MeasurableSpace α
μ : Measure α
inst✝² : TopologicalSpace β
f✝ g✝ : α → β
inst✝¹ : AddGroup β
inst✝ : TopologicalAddGroup β
f g : α → β
hfm : AEStronglyMeasurable' m f μ
hgm : AEStronglyMeasurable' m g μ
⊢ AEStronglyMeasurable' m (f - g) μ
state_after: case intro.intro
α : Type u_1
β : Type u_2
𝕜 : Type u_3
m m0 : MeasurableSpace α
μ : Measure α
inst✝² : TopologicalSpace β
f✝ g✝ : α → β
inst✝¹ : AddGroup β
inst✝ : TopologicalAddGroup β
f g : α → β
hgm : AEStronglyMeasurable' m g μ
f' : α → β
hf'_meas : StronglyMeasurable f'
hf_ae : f =ᶠ[ae μ] f'
⊢ AEStronglyMeasurable' m (f - g) μ
tactic: rcases hfm with ⟨f', hf'_meas, hf_ae⟩
state_before: case intro.intro
α : Type u_1
β : Type u_2
𝕜 : Type u_3
m m0 : MeasurableSpace α
μ : Measure α
inst✝² : TopologicalSpace β
f✝ g✝ : α → β
inst✝¹ : AddGroup β
inst✝ : TopologicalAddGroup β
f g : α → β
hgm : AEStronglyMeasurable' m g μ
f' : α → β
hf'_meas : StronglyMeasurable f'
hf_ae : f =ᶠ[ae μ] f'
⊢ AEStronglyMeasurable' m (f - g) μ
state_after: case intro.intro.intro.intro
α : Type u_1
β : Type u_2
𝕜 : Type u_3
m m0 : MeasurableSpace α
μ : Measure α
inst✝² : TopologicalSpace β
f✝ g✝ : α → β
inst✝¹ : AddGroup β
inst✝ : TopologicalAddGroup β
f g f' : α → β
hf'_meas : StronglyMeasurable f'
hf_ae : f =ᶠ[ae μ] f'
g' : α → β
hg'_meas : StronglyMeasurable g'
hg_ae : g =ᶠ[ae μ] g'
⊢ AEStronglyMeasurable' m (f - g) μ
tactic: rcases hgm with ⟨g', hg'_meas, hg_ae⟩
state_before: case intro.intro.intro.intro
α : Type u_1
β : Type u_2
𝕜 : Type u_3
m m0 : MeasurableSpace α
μ : Measure α
inst✝² : TopologicalSpace β
f✝ g✝ : α → β
inst✝¹ : AddGroup β
inst✝ : TopologicalAddGroup β
f g f' : α → β
hf'_meas : StronglyMeasurable f'
hf_ae : f =ᶠ[ae μ] f'
g' : α → β
hg'_meas : StronglyMeasurable g'
hg_ae : g =ᶠ[ae μ] g'
⊢ AEStronglyMeasurable' m (f - g) μ
state_after: case intro.intro.intro.intro
α : Type u_1
β : Type u_2
𝕜 : Type u_3
m m0 : MeasurableSpace α
μ : Measure α
inst✝² : TopologicalSpace β
f✝ g✝ : α → β
inst✝¹ : AddGroup β
inst✝ : TopologicalAddGroup β
f g f' : α → β
hf'_meas : StronglyMeasurable f'
hf_ae : f =ᶠ[ae μ] f'
g' : α → β
hg'_meas : StronglyMeasurable g'
hg_ae : g =ᶠ[ae μ] g'
x : α
hx1 : g x = g' x
hx2 : f x = f' x
⊢ (f - g) x = (f' - g') x
tactic: refine ⟨f' - g', hf'_meas.sub hg'_meas, hf_ae.mp (hg_ae.mono fun x hx1 hx2 => ?_)⟩
state_before: case intro.intro.intro.intro
α : Type u_1
β : Type u_2
𝕜 : Type u_3
m m0 : MeasurableSpace α
μ : Measure α
inst✝² : TopologicalSpace β
f✝ g✝ : α → β
inst✝¹ : AddGroup β
inst✝ : TopologicalAddGroup β
f g f' : α → β
hf'_meas : StronglyMeasurable f'
hf_ae : f =ᶠ[ae μ] f'
g' : α → β
hg'_meas : StronglyMeasurable g'
hg_ae : g =ᶠ[ae μ] g'
x : α
hx1 : g x = g' x
hx2 : f x = f' x
⊢ (f - g) x = (f' - g') x
state_after: case intro.intro.intro.intro
α : Type u_1
β : Type u_2
𝕜 : Type u_3
m m0 : MeasurableSpace α
μ : Measure α
inst✝² : TopologicalSpace β
f✝ g✝ : α → β
inst✝¹ : AddGroup β
inst✝ : TopologicalAddGroup β
f g f' : α → β
hf'_meas : StronglyMeasurable f'
hf_ae : f =ᶠ[ae μ] f'
g' : α → β
hg'_meas : StronglyMeasurable g'
hg_ae : g =ᶠ[ae μ] g'
x : α
hx1 : g x = g' x
hx2 : f x = f' x
⊢ f x - g x = f' x - g' x
tactic: simp_rw [Pi.sub_apply]
state_before: case intro.intro.intro.intro
α : Type u_1
β : Type u_2
𝕜 : Type u_3
m m0 : MeasurableSpace α
μ : Measure α
inst✝² : TopologicalSpace β
f✝ g✝ : α → β
inst✝¹ : AddGroup β
inst✝ : TopologicalAddGroup β
f g f' : α → β
hf'_meas : StronglyMeasurable f'
hf_ae : f =ᶠ[ae μ] f'
g' : α → β
hg'_meas : StronglyMeasurable g'
hg_ae : g =ᶠ[ae μ] g'
x : α
hx1 : g x = g' x
hx2 : f x = f' x
⊢ f x - g x = f' x - g' x
state_after: no goals
tactic: rw [hx1, hx2]