Theorem: MeasureTheory.SignedMeasure.eq_rnDeriv
File: Mathlib/MeasureTheory/Decomposition/SignedLebesgue.lean

Proof steps:
state_before: α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ ν : Measure α
s t✝ t : SignedMeasure α
f : α → ℝ
hfi : Integrable f μ
htμ : t ⟂ᵥ μ.toENNRealVectorMeasure
hadd : s = t + μ.withDensityᵥ f
⊢ f =ᶠ[ae μ] s.rnDeriv μ
state_after: α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ ν : Measure α
s t✝ t : SignedMeasure α
f : α → ℝ
hfi : Integrable f μ
htμ : t ⟂ᵥ μ.toENNRealVectorMeasure
hadd : s = t + μ.withDensityᵥ f
f' : α → ℝ := AEStronglyMeasurable.mk f ⋯
⊢ f =ᶠ[ae μ] s.rnDeriv μ
tactic: set f' := hfi.1.mk f
state_before: α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ ν : Measure α
s t✝ t : SignedMeasure α
f : α → ℝ
hfi : Integrable f μ
htμ : t ⟂ᵥ μ.toENNRealVectorMeasure
hadd : s = t + μ.withDensityᵥ f
f' : α → ℝ := AEStronglyMeasurable.mk f ⋯
⊢ f =ᶠ[ae μ] s.rnDeriv μ
state_after: α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ ν : Measure α
s t✝ t : SignedMeasure α
f : α → ℝ
hfi : Integrable f μ
htμ : t ⟂ᵥ μ.toENNRealVectorMeasure
hadd : s = t + μ.withDensityᵥ f
f' : α → ℝ := AEStronglyMeasurable.mk f ⋯
hadd' : s = t + μ.withDensityᵥ f'
⊢ f =ᶠ[ae μ] s.rnDeriv μ
tactic: have hadd' : s = t + μ.withDensityᵥ f' := by
    convert hadd using 2
    exact WithDensityᵥEq.congr_ae hfi.1.ae_eq_mk.symm
state_before: α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ ν : Measure α
s t✝ t : SignedMeasure α
f : α → ℝ
hfi : Integrable f μ
htμ : t ⟂ᵥ μ.toENNRealVectorMeasure
hadd : s = t + μ.withDensityᵥ f
f' : α → ℝ := AEStronglyMeasurable.mk f ⋯
hadd' : s = t + μ.withDensityᵥ f'
⊢ f =ᶠ[ae μ] s.rnDeriv μ
state_after: α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ ν : Measure α
s t✝ t : SignedMeasure α
f : α → ℝ
hfi : Integrable f μ
htμ : t ⟂ᵥ μ.toENNRealVectorMeasure
hadd : s = t + μ.withDensityᵥ f
f' : α → ℝ := AEStronglyMeasurable.mk f ⋯
hadd' : s = t + μ.withDensityᵥ f'
this : s.HaveLebesgueDecomposition μ
⊢ f =ᶠ[ae μ] s.rnDeriv μ
tactic: have := haveLebesgueDecomposition_mk μ hfi.1.measurable_mk htμ hadd'
state_before: α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ ν : Measure α
s t✝ t : SignedMeasure α
f : α → ℝ
hfi : Integrable f μ
htμ : t ⟂ᵥ μ.toENNRealVectorMeasure
hadd : s = t + μ.withDensityᵥ f
f' : α → ℝ := AEStronglyMeasurable.mk f ⋯
hadd' : s = t + μ.withDensityᵥ f'
this : s.HaveLebesgueDecomposition μ
⊢ f =ᶠ[ae μ] s.rnDeriv μ
state_after: α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ ν : Measure α
s t✝ t : SignedMeasure α
f : α → ℝ
hfi : Integrable f μ
htμ : t ⟂ᵥ μ.toENNRealVectorMeasure
hadd : s = t + μ.withDensityᵥ f
f' : α → ℝ := AEStronglyMeasurable.mk f ⋯
hadd' : s = t + μ.withDensityᵥ f'
this : s.HaveLebesgueDecomposition μ
⊢ μ.withDensityᵥ (s.rnDeriv μ) = μ.withDensityᵥ f
tactic: refine (Integrable.ae_eq_of_withDensityᵥ_eq (integrable_rnDeriv _ _) hfi ?_).symm
state_before: α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ ν : Measure α
s t✝ t : SignedMeasure α
f : α → ℝ
hfi : Integrable f μ
htμ : t ⟂ᵥ μ.toENNRealVectorMeasure
hadd : s = t + μ.withDensityᵥ f
f' : α → ℝ := AEStronglyMeasurable.mk f ⋯
hadd' : s = t + μ.withDensityᵥ f'
this : s.HaveLebesgueDecomposition μ
⊢ μ.withDensityᵥ (s.rnDeriv μ) = μ.withDensityᵥ f
state_after: no goals
tactic: rw [← add_right_inj t, ← hadd, eq_singularPart _ f htμ hadd,
    singularPart_add_withDensity_rnDeriv_eq]