[GOAL]
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
inst✝³ : MeasurableSpace α
μ ν : Measure α
inst✝² : TopologicalSpace β
inst✝¹ : TopologicalSpace γ
inst✝ : TopologicalSpace δ
f : α →ₘ[μ] β
⊢ mk ↑f (_ : AEStronglyMeasurable (↑f) μ) = f
[PROOFSTEP]
conv_rhs => rw [← Quotient.out_eq' f]
[GOAL]
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
inst✝³ : MeasurableSpace α
μ ν : Measure α
inst✝² : TopologicalSpace β
inst✝¹ : TopologicalSpace γ
inst✝ : TopologicalSpace δ
f : α →ₘ[μ] β
| f
[PROOFSTEP]
rw [← Quotient.out_eq' f]
[GOAL]
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
inst✝³ : MeasurableSpace α
μ ν : Measure α
inst✝² : TopologicalSpace β
inst✝¹ : TopologicalSpace γ
inst✝ : TopologicalSpace δ
f : α →ₘ[μ] β
| f
[PROOFSTEP]
rw [← Quotient.out_eq' f]
[GOAL]
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
inst✝³ : MeasurableSpace α
μ ν : Measure α
inst✝² : TopologicalSpace β
inst✝¹ : TopologicalSpace γ
inst✝ : TopologicalSpace δ
f : α →ₘ[μ] β
| f
[PROOFSTEP]
rw [← Quotient.out_eq' f]
[GOAL]
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
inst✝³ : MeasurableSpace α
μ ν : Measure α
inst✝² : TopologicalSpace β
inst✝¹ : TopologicalSpace γ
inst✝ : TopologicalSpace δ
f : α →ₘ[μ] β
⊢ mk ↑f (_ : AEStronglyMeasurable (↑f) μ) = Quotient.mk'' (Quotient.out' f)
[PROOFSTEP]
set g : { f : α → β // AEStronglyMeasurable f μ } := Quotient.out' f
[GOAL]
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
inst✝³ : MeasurableSpace α
μ ν : Measure α
inst✝² : TopologicalSpace β
inst✝¹ : TopologicalSpace γ
inst✝ : TopologicalSpace δ
f : α →ₘ[μ] β
g : { f // AEStronglyMeasurable f μ } := Quotient.out' f
⊢ mk ↑f (_ : AEStronglyMeasurable (↑f) μ) = Quotient.mk'' g
[PROOFSTEP]
have : g = ⟨g.1, g.2⟩ := Subtype.eq rfl
[GOAL]
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
inst✝³ : MeasurableSpace α
μ ν : Measure α
inst✝² : TopologicalSpace β
inst✝¹ : TopologicalSpace γ
inst✝ : TopologicalSpace δ
f : α →ₘ[μ] β
g : { f // AEStronglyMeasurable f μ } := Quotient.out' f
this : g = { val := ↑g, property := (_ : AEStronglyMeasurable (↑g) μ) }
⊢ mk ↑f (_ : AEStronglyMeasurable (↑f) μ) = Quotient.mk'' g
[PROOFSTEP]
rw [this, ← mk, mk_eq_mk]
[GOAL]
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
inst✝³ : MeasurableSpace α
μ ν : Measure α
inst✝² : TopologicalSpace β
inst✝¹ : TopologicalSpace γ
inst✝ : TopologicalSpace δ
f : α →ₘ[μ] β
g : { f // AEStronglyMeasurable f μ } := Quotient.out' f
this : g = { val := ↑g, property := (_ : AEStronglyMeasurable (↑g) μ) }
⊢ ↑f =ᵐ[μ] ↑g
[PROOFSTEP]
exact (AEStronglyMeasurable.ae_eq_mk _).symm
[GOAL]
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
inst✝³ : MeasurableSpace α
μ ν : Measure α
inst✝² : TopologicalSpace β
inst✝¹ : TopologicalSpace γ
inst✝ : TopologicalSpace δ
f g : α →ₘ[μ] β
h : ↑f =ᵐ[μ] ↑g
⊢ f = g
[PROOFSTEP]
rwa [← f.mk_coeFn, ← g.mk_coeFn, mk_eq_mk]
[GOAL]
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
inst✝³ : MeasurableSpace α
μ ν : Measure α
inst✝² : TopologicalSpace β
inst✝¹ : TopologicalSpace γ
inst✝ : TopologicalSpace δ
f g : α →ₘ[μ] β
h : f = g
⊢ ↑f =ᵐ[μ] ↑g
[PROOFSTEP]
rw [h]
[GOAL]
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
inst✝³ : MeasurableSpace α
μ ν : Measure α
inst✝² : TopologicalSpace β
inst✝¹ : TopologicalSpace γ
inst✝ : TopologicalSpace δ
f : α → β
hf : AEStronglyMeasurable f μ
⊢ ↑(mk f hf) =ᵐ[μ] f
[PROOFSTEP]
apply (AEStronglyMeasurable.ae_eq_mk _).symm.trans
[GOAL]
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
inst✝³ : MeasurableSpace α
μ ν : Measure α
inst✝² : TopologicalSpace β
inst✝¹ : TopologicalSpace γ
inst✝ : TopologicalSpace δ
f : α → β
hf : AEStronglyMeasurable f μ
⊢ ↑(Quotient.out' (mk f hf)) =ᵐ[μ] f
[PROOFSTEP]
exact @Quotient.mk_out' _ (μ.aeEqSetoid β) (⟨f, hf⟩ : { f // AEStronglyMeasurable f μ })
[GOAL]
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
inst✝⁴ : MeasurableSpace α
μ ν✝ : Measure α
inst✝³ : TopologicalSpace β
inst✝² : TopologicalSpace γ
inst✝¹ : TopologicalSpace δ
inst✝ : MeasurableSpace β
ν : Measure β
f : α → β
g : β →ₘ[ν] γ
hf : QuasiMeasurePreserving f
⊢ compQuasiMeasurePreserving g f hf = mk (↑g ∘ f) (_ : AEStronglyMeasurable (↑g ∘ f) μ)
[PROOFSTEP]
rw [← compQuasiMeasurePreserving_mk g.aestronglyMeasurable hf, mk_coeFn]
[GOAL]
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
inst✝⁴ : MeasurableSpace α
μ ν✝ : Measure α
inst✝³ : TopologicalSpace β
inst✝² : TopologicalSpace γ
inst✝¹ : TopologicalSpace δ
inst✝ : MeasurableSpace β
ν : Measure β
f : α → β
g : β →ₘ[ν] γ
hf : QuasiMeasurePreserving f
⊢ ↑(compQuasiMeasurePreserving g f hf) =ᵐ[μ] ↑g ∘ f
[PROOFSTEP]
rw [compQuasiMeasurePreserving_eq_mk]
[GOAL]
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
inst✝⁴ : MeasurableSpace α
μ ν✝ : Measure α
inst✝³ : TopologicalSpace β
inst✝² : TopologicalSpace γ
inst✝¹ : TopologicalSpace δ
inst✝ : MeasurableSpace β
ν : Measure β
f : α → β
g : β →ₘ[ν] γ
hf : QuasiMeasurePreserving f
⊢ ↑(mk (↑g ∘ f) (_ : AEStronglyMeasurable (↑g ∘ f) μ)) =ᵐ[μ] ↑g ∘ f
[PROOFSTEP]
apply coeFn_mk
[GOAL]
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
inst✝³ : MeasurableSpace α
μ ν : Measure α
inst✝² : TopologicalSpace β
inst✝¹ : TopologicalSpace γ
inst✝ : TopologicalSpace δ
g : β → γ
hg : Continuous g
f : α →ₘ[μ] β
⊢ comp g hg f = mk (g ∘ ↑f) (_ : AEStronglyMeasurable (fun x => g (↑f x)) μ)
[PROOFSTEP]
rw [← comp_mk g hg f f.aestronglyMeasurable, mk_coeFn]
[GOAL]
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
inst✝³ : MeasurableSpace α
μ ν : Measure α
inst✝² : TopologicalSpace β
inst✝¹ : TopologicalSpace γ
inst✝ : TopologicalSpace δ
g : β → γ
hg : Continuous g
f : α →ₘ[μ] β
⊢ ↑(comp g hg f) =ᵐ[μ] g ∘ ↑f
[PROOFSTEP]
rw [comp_eq_mk]
[GOAL]
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
inst✝³ : MeasurableSpace α
μ ν : Measure α
inst✝² : TopologicalSpace β
inst✝¹ : TopologicalSpace γ
inst✝ : TopologicalSpace δ
g : β → γ
hg : Continuous g
f : α →ₘ[μ] β
⊢ ↑(mk (g ∘ ↑f) (_ : AEStronglyMeasurable (fun x => g (↑f x)) μ)) =ᵐ[μ] g ∘ ↑f
[PROOFSTEP]
apply coeFn_mk
[GOAL]
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
inst✝¹⁰ : MeasurableSpace α
μ ν : Measure α
inst✝⁹ : TopologicalSpace β
inst✝⁸ : TopologicalSpace γ
inst✝⁷ : TopologicalSpace δ
inst✝⁶ : MeasurableSpace β
inst✝⁵ : PseudoMetrizableSpace β
inst✝⁴ : BorelSpace β
inst✝³ : MeasurableSpace γ
inst✝² : PseudoMetrizableSpace γ
inst✝¹ : OpensMeasurableSpace γ
inst✝ : SecondCountableTopology γ
g : β → γ
hg : Measurable g
f : α →ₘ[μ] β
⊢ compMeasurable g hg f = mk (g ∘ ↑f) (_ : AEStronglyMeasurable (g ∘ ↑f) μ)
[PROOFSTEP]
rw [← compMeasurable_mk g hg f f.aestronglyMeasurable, mk_coeFn]
[GOAL]
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
inst✝¹⁰ : MeasurableSpace α
μ ν : Measure α
inst✝⁹ : TopologicalSpace β
inst✝⁸ : TopologicalSpace γ
inst✝⁷ : TopologicalSpace δ
inst✝⁶ : MeasurableSpace β
inst✝⁵ : PseudoMetrizableSpace β
inst✝⁴ : BorelSpace β
inst✝³ : MeasurableSpace γ
inst✝² : PseudoMetrizableSpace γ
inst✝¹ : OpensMeasurableSpace γ
inst✝ : SecondCountableTopology γ
g : β → γ
hg : Measurable g
f : α →ₘ[μ] β
⊢ ↑(compMeasurable g hg f) =ᵐ[μ] g ∘ ↑f
[PROOFSTEP]
rw [compMeasurable_eq_mk]
[GOAL]
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
inst✝¹⁰ : MeasurableSpace α
μ ν : Measure α
inst✝⁹ : TopologicalSpace β
inst✝⁸ : TopologicalSpace γ
inst✝⁷ : TopologicalSpace δ
inst✝⁶ : MeasurableSpace β
inst✝⁵ : PseudoMetrizableSpace β
inst✝⁴ : BorelSpace β
inst✝³ : MeasurableSpace γ
inst✝² : PseudoMetrizableSpace γ
inst✝¹ : OpensMeasurableSpace γ
inst✝ : SecondCountableTopology γ
g : β → γ
hg : Measurable g
f : α →ₘ[μ] β
⊢ ↑(mk (g ∘ ↑f) (_ : AEStronglyMeasurable (g ∘ ↑f) μ)) =ᵐ[μ] g ∘ ↑f
[PROOFSTEP]
apply coeFn_mk
[GOAL]
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
inst✝³ : MeasurableSpace α
μ ν : Measure α
inst✝² : TopologicalSpace β
inst✝¹ : TopologicalSpace γ
inst✝ : TopologicalSpace δ
f : α →ₘ[μ] β
g : α →ₘ[μ] γ
⊢ pair f g = mk (fun x => (↑f x, ↑g x)) (_ : AEStronglyMeasurable (fun x => (↑f x, ↑g x)) μ)
[PROOFSTEP]
simp only [← pair_mk_mk, mk_coeFn, f.aestronglyMeasurable, g.aestronglyMeasurable]
[GOAL]
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
inst✝³ : MeasurableSpace α
μ ν : Measure α
inst✝² : TopologicalSpace β
inst✝¹ : TopologicalSpace γ
inst✝ : TopologicalSpace δ
f : α →ₘ[μ] β
g : α →ₘ[μ] γ
⊢ ↑(pair f g) =ᵐ[μ] fun x => (↑f x, ↑g x)
[PROOFSTEP]
rw [pair_eq_mk]
[GOAL]
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
inst✝³ : MeasurableSpace α
μ ν : Measure α
inst✝² : TopologicalSpace β
inst✝¹ : TopologicalSpace γ
inst✝ : TopologicalSpace δ
f : α →ₘ[μ] β
g : α →ₘ[μ] γ
⊢ ↑(mk (fun x => (↑f x, ↑g x)) (_ : AEStronglyMeasurable (fun x => (↑f x, ↑g x)) μ)) =ᵐ[μ] fun x => (↑f x, ↑g x)
[PROOFSTEP]
apply coeFn_mk
[GOAL]
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
inst✝³ : MeasurableSpace α
μ ν : Measure α
inst✝² : TopologicalSpace β
inst✝¹ : TopologicalSpace γ
inst✝ : TopologicalSpace δ
g : β → γ → δ
hg : Continuous (uncurry g)
f₁ : α →ₘ[μ] β
f₂ : α →ₘ[μ] γ
⊢ comp₂ g hg f₁ f₂ = mk (fun a => g (↑f₁ a) (↑f₂ a)) (_ : AEStronglyMeasurable (fun x => uncurry g (↑f₁ x, ↑f₂ x)) μ)
[PROOFSTEP]
rw [comp₂_eq_pair, pair_eq_mk, comp_mk]
[GOAL]
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
inst✝³ : MeasurableSpace α
μ ν : Measure α
inst✝² : TopologicalSpace β
inst✝¹ : TopologicalSpace γ
inst✝ : TopologicalSpace δ
g : β → γ → δ
hg : Continuous (uncurry g)
f₁ : α →ₘ[μ] β
f₂ : α →ₘ[μ] γ
⊢ mk (uncurry g ∘ fun x => (↑f₁ x, ↑f₂ x)) (_ : AEStronglyMeasurable (fun x => uncurry g (↑f₁ x, ↑f₂ x)) μ) =
    mk (fun a => g (↑f₁ a) (↑f₂ a)) (_ : AEStronglyMeasurable (fun x => uncurry g (↑f₁ x, ↑f₂ x)) μ)
[PROOFSTEP]
rfl
[GOAL]
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
inst✝³ : MeasurableSpace α
μ ν : Measure α
inst✝² : TopologicalSpace β
inst✝¹ : TopologicalSpace γ
inst✝ : TopologicalSpace δ
g : β → γ → δ
hg : Continuous (uncurry g)
f₁ : α →ₘ[μ] β
f₂ : α →ₘ[μ] γ
⊢ ↑(comp₂ g hg f₁ f₂) =ᵐ[μ] fun a => g (↑f₁ a) (↑f₂ a)
[PROOFSTEP]
rw [comp₂_eq_mk]
[GOAL]
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
inst✝³ : MeasurableSpace α
μ ν : Measure α
inst✝² : TopologicalSpace β
inst✝¹ : TopologicalSpace γ
inst✝ : TopologicalSpace δ
g : β → γ → δ
hg : Continuous (uncurry g)
f₁ : α →ₘ[μ] β
f₂ : α →ₘ[μ] γ
⊢ ↑(mk (fun a => g (↑f₁ a) (↑f₂ a)) (_ : AEStronglyMeasurable (fun x => uncurry g (↑f₁ x, ↑f₂ x)) μ)) =ᵐ[μ] fun a =>
    g (↑f₁ a) (↑f₂ a)
[PROOFSTEP]
apply coeFn_mk
[GOAL]
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
inst✝¹⁵ : MeasurableSpace α
μ ν : Measure α
inst✝¹⁴ : TopologicalSpace β
inst✝¹³ : TopologicalSpace γ
inst✝¹² : TopologicalSpace δ
inst✝¹¹ : MeasurableSpace β
inst✝¹⁰ : PseudoMetrizableSpace β
inst✝⁹ : BorelSpace β
inst✝⁸ : SecondCountableTopology β
inst✝⁷ : MeasurableSpace γ
inst✝⁶ : PseudoMetrizableSpace γ
inst✝⁵ : BorelSpace γ
inst✝⁴ : SecondCountableTopology γ
inst✝³ : MeasurableSpace δ
inst✝² : PseudoMetrizableSpace δ
inst✝¹ : OpensMeasurableSpace δ
inst✝ : SecondCountableTopology δ
g : β → γ → δ
hg : Measurable (uncurry g)
f₁ : α →ₘ[μ] β
f₂ : α →ₘ[μ] γ
⊢ comp₂Measurable g hg f₁ f₂ =
    mk (fun a => g (↑f₁ a) (↑f₂ a)) (_ : AEStronglyMeasurable (uncurry g ∘ fun x => (↑f₁ x, ↑f₂ x)) μ)
[PROOFSTEP]
rw [comp₂Measurable_eq_pair, pair_eq_mk, compMeasurable_mk]
[GOAL]
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
inst✝¹⁵ : MeasurableSpace α
μ ν : Measure α
inst✝¹⁴ : TopologicalSpace β
inst✝¹³ : TopologicalSpace γ
inst✝¹² : TopologicalSpace δ
inst✝¹¹ : MeasurableSpace β
inst✝¹⁰ : PseudoMetrizableSpace β
inst✝⁹ : BorelSpace β
inst✝⁸ : SecondCountableTopology β
inst✝⁷ : MeasurableSpace γ
inst✝⁶ : PseudoMetrizableSpace γ
inst✝⁵ : BorelSpace γ
inst✝⁴ : SecondCountableTopology γ
inst✝³ : MeasurableSpace δ
inst✝² : PseudoMetrizableSpace δ
inst✝¹ : OpensMeasurableSpace δ
inst✝ : SecondCountableTopology δ
g : β → γ → δ
hg : Measurable (uncurry g)
f₁ : α →ₘ[μ] β
f₂ : α →ₘ[μ] γ
⊢ mk (uncurry g ∘ fun x => (↑f₁ x, ↑f₂ x)) (_ : AEStronglyMeasurable (uncurry g ∘ fun x => (↑f₁ x, ↑f₂ x)) μ) =
    mk (fun a => g (↑f₁ a) (↑f₂ a)) (_ : AEStronglyMeasurable (uncurry g ∘ fun x => (↑f₁ x, ↑f₂ x)) μ)
[PROOFSTEP]
rfl
[GOAL]
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
inst✝¹⁵ : MeasurableSpace α
μ ν : Measure α
inst✝¹⁴ : TopologicalSpace β
inst✝¹³ : TopologicalSpace γ
inst✝¹² : TopologicalSpace δ
inst✝¹¹ : MeasurableSpace β
inst✝¹⁰ : PseudoMetrizableSpace β
inst✝⁹ : BorelSpace β
inst✝⁸ : SecondCountableTopology β
inst✝⁷ : MeasurableSpace γ
inst✝⁶ : PseudoMetrizableSpace γ
inst✝⁵ : BorelSpace γ
inst✝⁴ : SecondCountableTopology γ
inst✝³ : MeasurableSpace δ
inst✝² : PseudoMetrizableSpace δ
inst✝¹ : OpensMeasurableSpace δ
inst✝ : SecondCountableTopology δ
g : β → γ → δ
hg : Measurable (uncurry g)
f₁ : α →ₘ[μ] β
f₂ : α →ₘ[μ] γ
⊢ ↑(comp₂Measurable g hg f₁ f₂) =ᵐ[μ] fun a => g (↑f₁ a) (↑f₂ a)
[PROOFSTEP]
rw [comp₂Measurable_eq_mk]
[GOAL]
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
inst✝¹⁵ : MeasurableSpace α
μ ν : Measure α
inst✝¹⁴ : TopologicalSpace β
inst✝¹³ : TopologicalSpace γ
inst✝¹² : TopologicalSpace δ
inst✝¹¹ : MeasurableSpace β
inst✝¹⁰ : PseudoMetrizableSpace β
inst✝⁹ : BorelSpace β
inst✝⁸ : SecondCountableTopology β
inst✝⁷ : MeasurableSpace γ
inst✝⁶ : PseudoMetrizableSpace γ
inst✝⁵ : BorelSpace γ
inst✝⁴ : SecondCountableTopology γ
inst✝³ : MeasurableSpace δ
inst✝² : PseudoMetrizableSpace δ
inst✝¹ : OpensMeasurableSpace δ
inst✝ : SecondCountableTopology δ
g : β → γ → δ
hg : Measurable (uncurry g)
f₁ : α →ₘ[μ] β
f₂ : α →ₘ[μ] γ
⊢ ↑(mk (fun a => g (↑f₁ a) (↑f₂ a)) (_ : AEStronglyMeasurable (uncurry g ∘ fun x => (↑f₁ x, ↑f₂ x)) μ)) =ᵐ[μ] fun a =>
    g (↑f₁ a) (↑f₂ a)
[PROOFSTEP]
apply coeFn_mk
[GOAL]
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
inst✝³ : MeasurableSpace α
μ ν : Measure α
inst✝² : TopologicalSpace β
inst✝¹ : TopologicalSpace γ
inst✝ : TopologicalSpace δ
f : α →ₘ[μ] β
⊢ toGerm f = ↑↑f
[PROOFSTEP]
rw [← mk_toGerm, mk_coeFn]
[GOAL]
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
inst✝³ : MeasurableSpace α
μ ν : Measure α
inst✝² : TopologicalSpace β
inst✝¹ : TopologicalSpace γ
inst✝ : TopologicalSpace δ
f g : α →ₘ[μ] β
H : toGerm f = toGerm g
⊢ ↑↑f = ↑↑g
[PROOFSTEP]
rwa [← toGerm_eq, ← toGerm_eq]
[GOAL]
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
inst✝³ : MeasurableSpace α
μ ν : Measure α
inst✝² : TopologicalSpace β
inst✝¹ : TopologicalSpace γ
inst✝ : TopologicalSpace δ
g : β → γ
hg : Continuous g
f✝ : α →ₘ[μ] β
f : α → β
x✝ : AEStronglyMeasurable f μ
⊢ toGerm (comp g hg (mk f x✝)) = Germ.map g (toGerm (mk f x✝))
[PROOFSTEP]
simp
[GOAL]
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
inst✝¹⁰ : MeasurableSpace α
μ ν : Measure α
inst✝⁹ : TopologicalSpace β
inst✝⁸ : TopologicalSpace γ
inst✝⁷ : TopologicalSpace δ
inst✝⁶ : MeasurableSpace β
inst✝⁵ : BorelSpace β
inst✝⁴ : PseudoMetrizableSpace β
inst✝³ : PseudoMetrizableSpace γ
inst✝² : SecondCountableTopology γ
inst✝¹ : MeasurableSpace γ
inst✝ : OpensMeasurableSpace γ
g : β → γ
hg : Measurable g
f✝ : α →ₘ[μ] β
f : α → β
x✝ : AEStronglyMeasurable f μ
⊢ toGerm (compMeasurable g hg (mk f x✝)) = Germ.map g (toGerm (mk f x✝))
[PROOFSTEP]
simp
[GOAL]
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
inst✝³ : MeasurableSpace α
μ ν : Measure α
inst✝² : TopologicalSpace β
inst✝¹ : TopologicalSpace γ
inst✝ : TopologicalSpace δ
g : β → γ → δ
hg : Continuous (uncurry g)
f₁✝ : α →ₘ[μ] β
f₂✝ : α →ₘ[μ] γ
f₁ : α → β
x✝¹ : AEStronglyMeasurable f₁ μ
f₂ : α → γ
x✝ : AEStronglyMeasurable f₂ μ
⊢ toGerm (comp₂ g hg (mk f₁ x✝¹) (mk f₂ x✝)) = Germ.map₂ g (toGerm (mk f₁ x✝¹)) (toGerm (mk f₂ x✝))
[PROOFSTEP]
simp
[GOAL]
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
inst✝¹⁵ : MeasurableSpace α
μ ν : Measure α
inst✝¹⁴ : TopologicalSpace β
inst✝¹³ : TopologicalSpace γ
inst✝¹² : TopologicalSpace δ
inst✝¹¹ : PseudoMetrizableSpace β
inst✝¹⁰ : SecondCountableTopology β
inst✝⁹ : MeasurableSpace β
inst✝⁸ : BorelSpace β
inst✝⁷ : PseudoMetrizableSpace γ
inst✝⁶ : SecondCountableTopology γ
inst✝⁵ : MeasurableSpace γ
inst✝⁴ : BorelSpace γ
inst✝³ : PseudoMetrizableSpace δ
inst✝² : SecondCountableTopology δ
inst✝¹ : MeasurableSpace δ
inst✝ : OpensMeasurableSpace δ
g : β → γ → δ
hg : Measurable (uncurry g)
f₁✝ : α →ₘ[μ] β
f₂✝ : α →ₘ[μ] γ
f₁ : α → β
x✝¹ : AEStronglyMeasurable f₁ μ
f₂ : α → γ
x✝ : AEStronglyMeasurable f₂ μ
⊢ toGerm (comp₂Measurable g hg (mk f₁ x✝¹) (mk f₂ x✝)) = Germ.map₂ g (toGerm (mk f₁ x✝¹)) (toGerm (mk f₂ x✝))
[PROOFSTEP]
simp
[GOAL]
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
inst✝³ : MeasurableSpace α
μ ν : Measure α
inst✝² : TopologicalSpace β
inst✝¹ : TopologicalSpace γ
inst✝ : TopologicalSpace δ
r : β → γ → Prop
f : α →ₘ[μ] β
g : α →ₘ[μ] γ
⊢ LiftRel r f g ↔ ∀ᵐ (a : α) ∂μ, r (↑f a) (↑g a)
[PROOFSTEP]
rw [← liftRel_mk_mk, mk_coeFn, mk_coeFn]
[GOAL]
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
inst✝⁵ : MeasurableSpace α
μ ν : Measure α
inst✝⁴ : TopologicalSpace β
inst✝³ : TopologicalSpace γ
inst✝² : TopologicalSpace δ
inst✝¹ : SemilatticeSup β
inst✝ : ContinuousSup β
f g : α →ₘ[μ] β
⊢ f ≤ f ⊔ g
[PROOFSTEP]
rw [← coeFn_le]
[GOAL]
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
inst✝⁵ : MeasurableSpace α
μ ν : Measure α
inst✝⁴ : TopologicalSpace β
inst✝³ : TopologicalSpace γ
inst✝² : TopologicalSpace δ
inst✝¹ : SemilatticeSup β
inst✝ : ContinuousSup β
f g : α →ₘ[μ] β
⊢ ↑f ≤ᵐ[μ] ↑(f ⊔ g)
[PROOFSTEP]
filter_upwards [coeFn_sup f g] with _ ha
[GOAL]
case h
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
inst✝⁵ : MeasurableSpace α
μ ν : Measure α
inst✝⁴ : TopologicalSpace β
inst✝³ : TopologicalSpace γ
inst✝² : TopologicalSpace δ
inst✝¹ : SemilatticeSup β
inst✝ : ContinuousSup β
f g : α →ₘ[μ] β
a✝ : α
ha : ↑(f ⊔ g) a✝ = ↑f a✝ ⊔ ↑g a✝
⊢ ↑f a✝ ≤ ↑(f ⊔ g) a✝
[PROOFSTEP]
rw [ha]
[GOAL]
case h
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
inst✝⁵ : MeasurableSpace α
μ ν : Measure α
inst✝⁴ : TopologicalSpace β
inst✝³ : TopologicalSpace γ
inst✝² : TopologicalSpace δ
inst✝¹ : SemilatticeSup β
inst✝ : ContinuousSup β
f g : α →ₘ[μ] β
a✝ : α
ha : ↑(f ⊔ g) a✝ = ↑f a✝ ⊔ ↑g a✝
⊢ ↑f a✝ ≤ ↑f a✝ ⊔ ↑g a✝
[PROOFSTEP]
exact le_sup_left
[GOAL]
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
inst✝⁵ : MeasurableSpace α
μ ν : Measure α
inst✝⁴ : TopologicalSpace β
inst✝³ : TopologicalSpace γ
inst✝² : TopologicalSpace δ
inst✝¹ : SemilatticeSup β
inst✝ : ContinuousSup β
f g : α →ₘ[μ] β
⊢ g ≤ f ⊔ g
[PROOFSTEP]
rw [← coeFn_le]
[GOAL]
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
inst✝⁵ : MeasurableSpace α
μ ν : Measure α
inst✝⁴ : TopologicalSpace β
inst✝³ : TopologicalSpace γ
inst✝² : TopologicalSpace δ
inst✝¹ : SemilatticeSup β
inst✝ : ContinuousSup β
f g : α →ₘ[μ] β
⊢ ↑g ≤ᵐ[μ] ↑(f ⊔ g)
[PROOFSTEP]
filter_upwards [coeFn_sup f g] with _ ha
[GOAL]
case h
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
inst✝⁵ : MeasurableSpace α
μ ν : Measure α
inst✝⁴ : TopologicalSpace β
inst✝³ : TopologicalSpace γ
inst✝² : TopologicalSpace δ
inst✝¹ : SemilatticeSup β
inst✝ : ContinuousSup β
f g : α →ₘ[μ] β
a✝ : α
ha : ↑(f ⊔ g) a✝ = ↑f a✝ ⊔ ↑g a✝
⊢ ↑g a✝ ≤ ↑(f ⊔ g) a✝
[PROOFSTEP]
rw [ha]
[GOAL]
case h
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
inst✝⁵ : MeasurableSpace α
μ ν : Measure α
inst✝⁴ : TopologicalSpace β
inst✝³ : TopologicalSpace γ
inst✝² : TopologicalSpace δ
inst✝¹ : SemilatticeSup β
inst✝ : ContinuousSup β
f g : α →ₘ[μ] β
a✝ : α
ha : ↑(f ⊔ g) a✝ = ↑f a✝ ⊔ ↑g a✝
⊢ ↑g a✝ ≤ ↑f a✝ ⊔ ↑g a✝
[PROOFSTEP]
exact le_sup_right
[GOAL]
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
inst✝⁵ : MeasurableSpace α
μ ν : Measure α
inst✝⁴ : TopologicalSpace β
inst✝³ : TopologicalSpace γ
inst✝² : TopologicalSpace δ
inst✝¹ : SemilatticeSup β
inst✝ : ContinuousSup β
f g f' : α →ₘ[μ] β
hf : f ≤ f'
hg : g ≤ f'
⊢ f ⊔ g ≤ f'
[PROOFSTEP]
rw [← coeFn_le] at hf hg ⊢
[GOAL]
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
inst✝⁵ : MeasurableSpace α
μ ν : Measure α
inst✝⁴ : TopologicalSpace β
inst✝³ : TopologicalSpace γ
inst✝² : TopologicalSpace δ
inst✝¹ : SemilatticeSup β
inst✝ : ContinuousSup β
f g f' : α →ₘ[μ] β
hf : ↑f ≤ᵐ[μ] ↑f'
hg : ↑g ≤ᵐ[μ] ↑f'
⊢ ↑(f ⊔ g) ≤ᵐ[μ] ↑f'
[PROOFSTEP]
filter_upwards [hf, hg, coeFn_sup f g] with _ haf hag ha_sup
[GOAL]
case h
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
inst✝⁵ : MeasurableSpace α
μ ν : Measure α
inst✝⁴ : TopologicalSpace β
inst✝³ : TopologicalSpace γ
inst✝² : TopologicalSpace δ
inst✝¹ : SemilatticeSup β
inst✝ : ContinuousSup β
f g f' : α →ₘ[μ] β
hf : ↑f ≤ᵐ[μ] ↑f'
hg : ↑g ≤ᵐ[μ] ↑f'
a✝ : α
haf : ↑f a✝ ≤ ↑f' a✝
hag : ↑g a✝ ≤ ↑f' a✝
ha_sup : ↑(f ⊔ g) a✝ = ↑f a✝ ⊔ ↑g a✝
⊢ ↑(f ⊔ g) a✝ ≤ ↑f' a✝
[PROOFSTEP]
rw [ha_sup]
[GOAL]
case h
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
inst✝⁵ : MeasurableSpace α
μ ν : Measure α
inst✝⁴ : TopologicalSpace β
inst✝³ : TopologicalSpace γ
inst✝² : TopologicalSpace δ
inst✝¹ : SemilatticeSup β
inst✝ : ContinuousSup β
f g f' : α →ₘ[μ] β
hf : ↑f ≤ᵐ[μ] ↑f'
hg : ↑g ≤ᵐ[μ] ↑f'
a✝ : α
haf : ↑f a✝ ≤ ↑f' a✝
hag : ↑g a✝ ≤ ↑f' a✝
ha_sup : ↑(f ⊔ g) a✝ = ↑f a✝ ⊔ ↑g a✝
⊢ ↑f a✝ ⊔ ↑g a✝ ≤ ↑f' a✝
[PROOFSTEP]
exact sup_le haf hag
[GOAL]
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
inst✝⁵ : MeasurableSpace α
μ ν : Measure α
inst✝⁴ : TopologicalSpace β
inst✝³ : TopologicalSpace γ
inst✝² : TopologicalSpace δ
inst✝¹ : SemilatticeInf β
inst✝ : ContinuousInf β
f g : α →ₘ[μ] β
⊢ f ⊓ g ≤ f
[PROOFSTEP]
rw [← coeFn_le]
[GOAL]
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
inst✝⁵ : MeasurableSpace α
μ ν : Measure α
inst✝⁴ : TopologicalSpace β
inst✝³ : TopologicalSpace γ
inst✝² : TopologicalSpace δ
inst✝¹ : SemilatticeInf β
inst✝ : ContinuousInf β
f g : α →ₘ[μ] β
⊢ ↑(f ⊓ g) ≤ᵐ[μ] ↑f
[PROOFSTEP]
filter_upwards [coeFn_inf f g] with _ ha
[GOAL]
case h
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
inst✝⁵ : MeasurableSpace α
μ ν : Measure α
inst✝⁴ : TopologicalSpace β
inst✝³ : TopologicalSpace γ
inst✝² : TopologicalSpace δ
inst✝¹ : SemilatticeInf β
inst✝ : ContinuousInf β
f g : α →ₘ[μ] β
a✝ : α
ha : ↑(f ⊓ g) a✝ = ↑f a✝ ⊓ ↑g a✝
⊢ ↑(f ⊓ g) a✝ ≤ ↑f a✝
[PROOFSTEP]
rw [ha]
[GOAL]
case h
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
inst✝⁵ : MeasurableSpace α
μ ν : Measure α
inst✝⁴ : TopologicalSpace β
inst✝³ : TopologicalSpace γ
inst✝² : TopologicalSpace δ
inst✝¹ : SemilatticeInf β
inst✝ : ContinuousInf β
f g : α →ₘ[μ] β
a✝ : α
ha : ↑(f ⊓ g) a✝ = ↑f a✝ ⊓ ↑g a✝
⊢ ↑f a✝ ⊓ ↑g a✝ ≤ ↑f a✝
[PROOFSTEP]
exact inf_le_left
[GOAL]
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
inst✝⁵ : MeasurableSpace α
μ ν : Measure α
inst✝⁴ : TopologicalSpace β
inst✝³ : TopologicalSpace γ
inst✝² : TopologicalSpace δ
inst✝¹ : SemilatticeInf β
inst✝ : ContinuousInf β
f g : α →ₘ[μ] β
⊢ f ⊓ g ≤ g
[PROOFSTEP]
rw [← coeFn_le]
[GOAL]
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
inst✝⁵ : MeasurableSpace α
μ ν : Measure α
inst✝⁴ : TopologicalSpace β
inst✝³ : TopologicalSpace γ
inst✝² : TopologicalSpace δ
inst✝¹ : SemilatticeInf β
inst✝ : ContinuousInf β
f g : α →ₘ[μ] β
⊢ ↑(f ⊓ g) ≤ᵐ[μ] ↑g
[PROOFSTEP]
filter_upwards [coeFn_inf f g] with _ ha
[GOAL]
case h
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
inst✝⁵ : MeasurableSpace α
μ ν : Measure α
inst✝⁴ : TopologicalSpace β
inst✝³ : TopologicalSpace γ
inst✝² : TopologicalSpace δ
inst✝¹ : SemilatticeInf β
inst✝ : ContinuousInf β
f g : α →ₘ[μ] β
a✝ : α
ha : ↑(f ⊓ g) a✝ = ↑f a✝ ⊓ ↑g a✝
⊢ ↑(f ⊓ g) a✝ ≤ ↑g a✝
[PROOFSTEP]
rw [ha]
[GOAL]
case h
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
inst✝⁵ : MeasurableSpace α
μ ν : Measure α
inst✝⁴ : TopologicalSpace β
inst✝³ : TopologicalSpace γ
inst✝² : TopologicalSpace δ
inst✝¹ : SemilatticeInf β
inst✝ : ContinuousInf β
f g : α →ₘ[μ] β
a✝ : α
ha : ↑(f ⊓ g) a✝ = ↑f a✝ ⊓ ↑g a✝
⊢ ↑f a✝ ⊓ ↑g a✝ ≤ ↑g a✝
[PROOFSTEP]
exact inf_le_right
[GOAL]
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
inst✝⁵ : MeasurableSpace α
μ ν : Measure α
inst✝⁴ : TopologicalSpace β
inst✝³ : TopologicalSpace γ
inst✝² : TopologicalSpace δ
inst✝¹ : SemilatticeInf β
inst✝ : ContinuousInf β
f' f g : α →ₘ[μ] β
hf : f' ≤ f
hg : f' ≤ g
⊢ f' ≤ f ⊓ g
[PROOFSTEP]
rw [← coeFn_le] at hf hg ⊢
[GOAL]
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
inst✝⁵ : MeasurableSpace α
μ ν : Measure α
inst✝⁴ : TopologicalSpace β
inst✝³ : TopologicalSpace γ
inst✝² : TopologicalSpace δ
inst✝¹ : SemilatticeInf β
inst✝ : ContinuousInf β
f' f g : α →ₘ[μ] β
hf : ↑f' ≤ᵐ[μ] ↑f
hg : ↑f' ≤ᵐ[μ] ↑g
⊢ ↑f' ≤ᵐ[μ] ↑(f ⊓ g)
[PROOFSTEP]
filter_upwards [hf, hg, coeFn_inf f g] with _ haf hag ha_inf
[GOAL]
case h
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
inst✝⁵ : MeasurableSpace α
μ ν : Measure α
inst✝⁴ : TopologicalSpace β
inst✝³ : TopologicalSpace γ
inst✝² : TopologicalSpace δ
inst✝¹ : SemilatticeInf β
inst✝ : ContinuousInf β
f' f g : α →ₘ[μ] β
hf : ↑f' ≤ᵐ[μ] ↑f
hg : ↑f' ≤ᵐ[μ] ↑g
a✝ : α
haf : ↑f' a✝ ≤ ↑f a✝
hag : ↑f' a✝ ≤ ↑g a✝
ha_inf : ↑(f ⊓ g) a✝ = ↑f a✝ ⊓ ↑g a✝
⊢ ↑f' a✝ ≤ ↑(f ⊓ g) a✝
[PROOFSTEP]
rw [ha_inf]
[GOAL]
case h
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
inst✝⁵ : MeasurableSpace α
μ ν : Measure α
inst✝⁴ : TopologicalSpace β
inst✝³ : TopologicalSpace γ
inst✝² : TopologicalSpace δ
inst✝¹ : SemilatticeInf β
inst✝ : ContinuousInf β
f' f g : α →ₘ[μ] β
hf : ↑f' ≤ᵐ[μ] ↑f
hg : ↑f' ≤ᵐ[μ] ↑g
a✝ : α
haf : ↑f' a✝ ≤ ↑f a✝
hag : ↑f' a✝ ≤ ↑g a✝
ha_inf : ↑(f ⊓ g) a✝ = ↑f a✝ ⊓ ↑g a✝
⊢ ↑f' a✝ ≤ ↑f a✝ ⊓ ↑g a✝
[PROOFSTEP]
exact le_inf haf hag
[GOAL]
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
inst✝⁸ : MeasurableSpace α
μ ν : Measure α
inst✝⁷ : TopologicalSpace β
inst✝⁶ : TopologicalSpace γ
inst✝⁵ : TopologicalSpace δ
𝕜 : Type u_5
𝕜' : Type u_6
inst✝⁴ : SMul 𝕜 γ
inst✝³ : ContinuousConstSMul 𝕜 γ
inst✝² : SMul 𝕜' γ
inst✝¹ : ContinuousConstSMul 𝕜' γ
inst✝ : SMulCommClass 𝕜 𝕜' γ
a : 𝕜
b : 𝕜'
f✝ : α →ₘ[μ] γ
f : α → γ
hf : AEStronglyMeasurable f μ
⊢ a • b • mk f hf = b • a • mk f hf
[PROOFSTEP]
simp_rw [smul_mk, smul_comm]
[GOAL]
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
inst✝⁹ : MeasurableSpace α
μ ν : Measure α
inst✝⁸ : TopologicalSpace β
inst✝⁷ : TopologicalSpace γ
inst✝⁶ : TopologicalSpace δ
𝕜 : Type u_5
𝕜' : Type u_6
inst✝⁵ : SMul 𝕜 γ
inst✝⁴ : ContinuousConstSMul 𝕜 γ
inst✝³ : SMul 𝕜' γ
inst✝² : ContinuousConstSMul 𝕜' γ
inst✝¹ : SMul 𝕜 𝕜'
inst✝ : IsScalarTower 𝕜 𝕜' γ
a : 𝕜
b : 𝕜'
f✝ : α →ₘ[μ] γ
f : α → γ
hf : AEStronglyMeasurable f μ
⊢ (a • b) • mk f hf = a • b • mk f hf
[PROOFSTEP]
simp_rw [smul_mk, smul_assoc]
[GOAL]
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
inst✝⁹ : MeasurableSpace α
μ ν : Measure α
inst✝⁸ : TopologicalSpace β
inst✝⁷ : TopologicalSpace γ
inst✝⁶ : TopologicalSpace δ
𝕜 : Type u_5
𝕜' : Type u_6
inst✝⁵ : SMul 𝕜 γ
inst✝⁴ : ContinuousConstSMul 𝕜 γ
inst✝³ : SMul 𝕜' γ
inst✝² : ContinuousConstSMul 𝕜' γ
inst✝¹ : SMul 𝕜ᵐᵒᵖ γ
inst✝ : IsCentralScalar 𝕜 γ
a : 𝕜
f✝ : α →ₘ[μ] γ
f : α → γ
hf : AEStronglyMeasurable f μ
⊢ MulOpposite.op a • mk f hf = a • mk f hf
[PROOFSTEP]
simp_rw [smul_mk, op_smul_eq_smul]
[GOAL]
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
inst✝³ : MeasurableSpace α
μ ν : Measure α
inst✝² : TopologicalSpace β
inst✝¹ : TopologicalSpace γ
inst✝ : TopologicalSpace δ
f : α →ₘ[μ] ℝ≥0∞
⊢ ∫⁻ (a : α), ↑f a ∂μ = lintegral f
[PROOFSTEP]
rw [← lintegral_mk, mk_coeFn]
[GOAL]
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
inst✝³ : MeasurableSpace α
μ ν : Measure α
inst✝² : TopologicalSpace β
inst✝¹ : TopologicalSpace γ
inst✝ : TopologicalSpace δ
f✝ g✝ : α →ₘ[μ] ℝ≥0∞
f : α → ℝ≥0∞
hf : AEStronglyMeasurable f μ
g : α → ℝ≥0∞
x✝ : AEStronglyMeasurable g μ
⊢ lintegral (mk f hf + mk g x✝) = lintegral (mk f hf) + lintegral (mk g x✝)
[PROOFSTEP]
simp [lintegral_add_left' hf.aemeasurable]
[GOAL]
α : Type u_1
β✝ : Type u_2
γ : Type u_3
δ : Type u_4
inst✝⁸ : MeasurableSpace α
μ ν : Measure α
inst✝⁷ : TopologicalSpace β✝
inst✝⁶ : TopologicalSpace γ
inst✝⁵ : TopologicalSpace δ
β : Type u_5
inst✝⁴ : TopologicalSpace β
inst✝³ : Lattice β
inst✝² : TopologicalLattice β
inst✝¹ : AddGroup β
inst✝ : TopologicalAddGroup β
f : α →ₘ[μ] β
⊢ ↑|f| =ᵐ[μ] fun x => |↑f x|
[PROOFSTEP]
simp_rw [abs_eq_sup_neg]
[GOAL]
α : Type u_1
β✝ : Type u_2
γ : Type u_3
δ : Type u_4
inst✝⁸ : MeasurableSpace α
μ ν : Measure α
inst✝⁷ : TopologicalSpace β✝
inst✝⁶ : TopologicalSpace γ
inst✝⁵ : TopologicalSpace δ
β : Type u_5
inst✝⁴ : TopologicalSpace β
inst✝³ : Lattice β
inst✝² : TopologicalLattice β
inst✝¹ : AddGroup β
inst✝ : TopologicalAddGroup β
f : α →ₘ[μ] β
⊢ ↑(f ⊔ -f) =ᵐ[μ] fun x => ↑f x ⊔ -↑f x
[PROOFSTEP]
filter_upwards [AEEqFun.coeFn_sup f (-f), AEEqFun.coeFn_neg f] with x hx_sup hx_neg
[GOAL]
case h
α : Type u_1
β✝ : Type u_2
γ : Type u_3
δ : Type u_4
inst✝⁸ : MeasurableSpace α
μ ν : Measure α
inst✝⁷ : TopologicalSpace β✝
inst✝⁶ : TopologicalSpace γ
inst✝⁵ : TopologicalSpace δ
β : Type u_5
inst✝⁴ : TopologicalSpace β
inst✝³ : Lattice β
inst✝² : TopologicalLattice β
inst✝¹ : AddGroup β
inst✝ : TopologicalAddGroup β
f : α →ₘ[μ] β
x : α
hx_sup : ↑(f ⊔ -f) x = ↑f x ⊔ ↑(-f) x
hx_neg : ↑(-f) x = (-↑f) x
⊢ ↑(f ⊔ -f) x = ↑f x ⊔ -↑f x
[PROOFSTEP]
rw [hx_sup, hx_neg, Pi.neg_apply]