Module: Mathlib.MeasureTheory.VectorMeasure.Decomposition.Lebesgue

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
μ : Measure α
s : SignedMeasure α
inst✝ : s.HaveLebesgueDecomposition μ
⊢ (s.toJordanDecomposition.posPart.singularPart μ +
          μ.withDensity (s.toJordanDecomposition.posPart.rnDeriv μ)).toSignedMeasure -
      (μ.withDensity (s.toJordanDecomposition.negPart.rnDeriv μ) +
          s.toJordanDecomposition.negPart.singularPart μ).toSignedMeasure =
    s.toJordanDecomposition.posPart.toSignedMeasure - s.toJordanDecomposition.negPart.toSignedMeasure

pp: case hfm
α : Type u_1
m : MeasurableSpace α
μ : Measure α
s : SignedMeasure α
inst✝ : s.HaveLebesgueDecomposition μ
⊢ AEMeasurable (s.toJordanDecomposition.negPart.rnDeriv μ) μ

pp: case hf
α : Type u_1
m : MeasurableSpace α
μ : Measure α
s : SignedMeasure α
inst✝ : s.HaveLebesgueDecomposition μ
⊢ ∫⁻ (x : α), s.toJordanDecomposition.negPart.rnDeriv μ x ∂μ ≠ ∞

pp: case hfm
α : Type u_1
m : MeasurableSpace α
μ : Measure α
s : SignedMeasure α
inst✝ : s.HaveLebesgueDecomposition μ
⊢ AEMeasurable (s.toJordanDecomposition.posPart.rnDeriv μ) μ

pp: case hf
α : Type u_1
m : MeasurableSpace α
μ : Measure α
s : SignedMeasure α
inst✝ : s.HaveLebesgueDecomposition μ
⊢ ∫⁻ (x : α), s.toJordanDecomposition.posPart.rnDeriv μ x ∂μ ≠ ∞

pp: α : Type u_1
m : MeasurableSpace α
μ : Measure α
s : SignedMeasure α
inst✝ : s.HaveLebesgueDecomposition μ
⊢ AEMeasurable (s.toJordanDecomposition.posPart.rnDeriv μ) μ

pp: α : Type u_1
m : MeasurableSpace α
μ : Measure α
s : SignedMeasure α
inst✝ : s.HaveLebesgueDecomposition μ
⊢ ∫⁻ (x : α), s.toJordanDecomposition.posPart.rnDeriv μ x ∂μ ≠ ∞

pp: α : Type u_1
m : MeasurableSpace α
μ : Measure α
s : SignedMeasure α
inst✝ : s.HaveLebesgueDecomposition μ
⊢ AEMeasurable (s.toJordanDecomposition.negPart.rnDeriv μ) μ

pp: α : Type u_1
m : MeasurableSpace α
μ : Measure α
s : SignedMeasure α
inst✝ : s.HaveLebesgueDecomposition μ
⊢ ∫⁻ (x : α), s.toJordanDecomposition.negPart.rnDeriv μ x ∂μ ≠ ∞

Tactic:
· convert! rfl
    -- `convert rfl` much faster than `congr`
    · exact s.toJordanDecomposition.posPart.haveLebesgueDecomposition_add μ
    · rw [add_comm]
      exact s.toJordanDecomposition.negPart.haveLebesgueDecomposition_add μ

Goal state after:
case hfm
α : Type u_1
m : MeasurableSpace α
μ : Measure α
s : SignedMeasure α
inst✝ : s.HaveLebesgueDecomposition μ
⊢ AEMeasurable (s.toJordanDecomposition.negPart.rnDeriv μ) μ

case hf
α : Type u_1
m : MeasurableSpace α
μ : Measure α
s : SignedMeasure α
inst✝ : s.HaveLebesgueDecomposition μ
⊢ ∫⁻ (x : α), s.toJordanDecomposition.negPart.rnDeriv μ x ∂μ ≠ ∞

case hfm
α : Type u_1
m : MeasurableSpace α
μ : Measure α
s : SignedMeasure α
inst✝ : s.HaveLebesgueDecomposition μ
⊢ AEMeasurable (s.toJordanDecomposition.posPart.rnDeriv μ) μ

case hf
α : Type u_1
m : MeasurableSpace α
μ : Measure α
s : SignedMeasure α
inst✝ : s.HaveLebesgueDecomposition μ
⊢ ∫⁻ (x : α), s.toJordanDecomposition.posPart.rnDeriv μ x ∂μ ≠ ∞

α : Type u_1
m : MeasurableSpace α
μ : Measure α
s : SignedMeasure α
inst✝ : s.HaveLebesgueDecomposition μ
⊢ AEMeasurable (s.toJordanDecomposition.posPart.rnDeriv μ) μ

α : Type u_1
m : MeasurableSpace α
μ : Measure α
s : SignedMeasure α
inst✝ : s.HaveLebesgueDecomposition μ
⊢ ∫⁻ (x : α), s.toJordanDecomposition.posPart.rnDeriv μ x ∂μ ≠ ∞

α : Type u_1
m : MeasurableSpace α
μ : Measure α
s : SignedMeasure α
inst✝ : s.HaveLebesgueDecomposition μ
⊢ AEMeasurable (s.toJordanDecomposition.negPart.rnDeriv μ) μ

α : Type u_1
m : MeasurableSpace α
μ : Measure α
s : SignedMeasure α
inst✝ : s.HaveLebesgueDecomposition μ
⊢ ∫⁻ (x : α), s.toJordanDecomposition.negPart.rnDeriv μ x ∂μ ≠ ∞