Theorem: MeasureTheory.SignedMeasure.singularPart_add_withDensity_rnDeriv_eq
File: Mathlib/MeasureTheory/Decomposition/Lebesgue.lean

tactic: conv_rhs =>
  rw [← toSignedMeasure_toJordanDecomposition s, JordanDecomposition.toSignedMeasure]
state_before: α : Type u_1
β : Type ?u.432299
m : MeasurableSpace α
μ ν : Measure α
s t : SignedMeasure α
inst✝ : HaveLebesgueDecomposition s μ
⊢ singularPart s μ + withDensityᵥ μ (rnDeriv s μ) = s

tactic: rw [singularPart, rnDeriv,
  withDensityᵥ_sub' (integrable_toReal_of_lintegral_ne_top _ _)
    (integrable_toReal_of_lintegral_ne_top _ _),
  withDensityᵥ_toReal, withDensityᵥ_toReal, sub_eq_add_neg, sub_eq_add_neg,
  add_comm (s.toJordanDecomposition.posPart.singularPart μ).toSignedMeasure, ← add_assoc,
  add_assoc (-(s.toJordanDecomposition.negPart.singularPart μ).toSignedMeasure),
  ← toSignedMeasure_add, add_comm, ← add_assoc, ← neg_add, ← toSignedMeasure_add, add_comm,
  ← sub_eq_add_neg]
state_before: α : Type u_1
β : Type ?u.432299
m : MeasurableSpace α
μ ν : Measure α
s t : SignedMeasure α
inst✝ : HaveLebesgueDecomposition s μ
⊢ singularPart s μ + withDensityᵥ μ (rnDeriv s μ) =
    toSignedMeasure (toJordanDecomposition s).posPart - toSignedMeasure (toJordanDecomposition s).negPart

tactic: convert rfl
state_before: α : Type u_1
β : Type ?u.432299
m : MeasurableSpace α
μ ν : Measure α
s t : SignedMeasure α
inst✝ : HaveLebesgueDecomposition s μ
⊢ toSignedMeasure
        (Measure.singularPart (toJordanDecomposition s).posPart μ +
          withDensity μ fun x => Measure.rnDeriv (toJordanDecomposition s).posPart μ x) -
      toSignedMeasure
        ((withDensity μ fun x => Measure.rnDeriv (toJordanDecomposition s).negPart μ x) +
          Measure.singularPart (toJordanDecomposition s).negPart μ) =
    toSignedMeasure (toJordanDecomposition s).posPart - toSignedMeasure (toJordanDecomposition s).negPart

case hf
α : Type u_1
β : Type ?u.432299
m : MeasurableSpace α
μ ν : Measure α
s t : SignedMeasure α
inst✝ : HaveLebesgueDecomposition s μ
⊢ (∫⁻ (x : α), Measure.rnDeriv (toJordanDecomposition s).posPart μ x ∂μ) ≠ ⊤

case hf
α : Type u_1
β : Type ?u.432299
m : MeasurableSpace α
μ ν : Measure α
s t : SignedMeasure α
inst✝ : HaveLebesgueDecomposition s μ
⊢ (∫⁻ (x : α), Measure.rnDeriv (toJordanDecomposition s).negPart μ x ∂μ) ≠ ⊤

case hf
α : Type u_1
β : Type ?u.432299
m : MeasurableSpace α
μ ν : Measure α
s t : SignedMeasure α
inst✝ : HaveLebesgueDecomposition s μ
⊢ (∫⁻ (x : α), Measure.rnDeriv (toJordanDecomposition s).negPart μ x ∂μ) ≠ ⊤

case hf
α : Type u_1
β : Type ?u.432299
m : MeasurableSpace α
μ ν : Measure α
s t : SignedMeasure α
inst✝ : HaveLebesgueDecomposition s μ
⊢ (∫⁻ (x : α), Measure.rnDeriv (toJordanDecomposition s).posPart μ x ∂μ) ≠ ⊤

case hf
α : Type u_1
β : Type ?u.432299
m : MeasurableSpace α
μ ν : Measure α
s t : SignedMeasure α
inst✝ : HaveLebesgueDecomposition s μ
⊢ (∫⁻ (x : α), Measure.rnDeriv (toJordanDecomposition s).negPart μ x ∂μ) ≠ ⊤

case hf
α : Type u_1
β : Type ?u.432299
m : MeasurableSpace α
μ ν : Measure α
s t : SignedMeasure α
inst✝ : HaveLebesgueDecomposition s μ
⊢ (∫⁻ (x : α), Measure.rnDeriv (toJordanDecomposition s).posPart μ x ∂μ) ≠ ⊤

case hf
α : Type u_1
β : Type ?u.432299
m : MeasurableSpace α
μ ν : Measure α
s t : SignedMeasure α
inst✝ : HaveLebesgueDecomposition s μ
⊢ (∫⁻ (x : α), Measure.rnDeriv (toJordanDecomposition s).negPart μ x ∂μ) ≠ ⊤

case hf
α : Type u_1
β : Type ?u.432299
m : MeasurableSpace α
μ ν : Measure α
s t : SignedMeasure α
inst✝ : HaveLebesgueDecomposition s μ
⊢ (∫⁻ (x : α), Measure.rnDeriv (toJordanDecomposition s).posPart μ x ∂μ) ≠ ⊤

case hf
α : Type u_1
β : Type ?u.432299
m : MeasurableSpace α
μ ν : Measure α
s t : SignedMeasure α
inst✝ : HaveLebesgueDecomposition s μ
⊢ (∫⁻ (x : α), Measure.rnDeriv (toJordanDecomposition s).negPart μ x ∂μ) ≠ ⊤

case hf
α : Type u_1
β : Type ?u.432299
m : MeasurableSpace α
μ ν : Measure α
s t : SignedMeasure α
inst✝ : HaveLebesgueDecomposition s μ
⊢ (∫⁻ (x : α), Measure.rnDeriv (toJordanDecomposition s).posPart μ x ∂μ) ≠ ⊤

case hf
α : Type u_1
β : Type ?u.432299
m : MeasurableSpace α
μ ν : Measure α
s t : SignedMeasure α
inst✝ : HaveLebesgueDecomposition s μ
⊢ (∫⁻ (x : α), Measure.rnDeriv (toJordanDecomposition s).posPart μ x ∂μ) ≠ ⊤

case hf
α : Type u_1
β : Type ?u.432299
m : MeasurableSpace α
μ ν : Measure α
s t : SignedMeasure α
inst✝ : HaveLebesgueDecomposition s μ
⊢ (∫⁻ (x : α), Measure.rnDeriv (toJordanDecomposition s).negPart μ x ∂μ) ≠ ⊤

case hf
α : Type u_1
β : Type ?u.432299
m : MeasurableSpace α
μ ν : Measure α
s t : SignedMeasure α
inst✝ : HaveLebesgueDecomposition s μ
⊢ (∫⁻ (x : α), Measure.rnDeriv (toJordanDecomposition s).posPart μ x ∂μ) ≠ ⊤

case hf
α : Type u_1
β : Type ?u.432299
m : MeasurableSpace α
μ ν : Measure α
s t : SignedMeasure α
inst✝ : HaveLebesgueDecomposition s μ
⊢ (∫⁻ (x : α), Measure.rnDeriv (toJordanDecomposition s).negPart μ x ∂μ) ≠ ⊤

case hf
α : Type u_1
β : Type ?u.432299
m : MeasurableSpace α
μ ν : Measure α
s t : SignedMeasure α
inst✝ : HaveLebesgueDecomposition s μ
⊢ (∫⁻ (x : α), Measure.rnDeriv (toJordanDecomposition s).posPart μ x ∂μ) ≠ ⊤

case hf
α : Type u_1
β : Type ?u.432299
m : MeasurableSpace α
μ ν : Measure α
s t : SignedMeasure α
inst✝ : HaveLebesgueDecomposition s μ
⊢ (∫⁻ (x : α), Measure.rnDeriv (toJordanDecomposition s).negPart μ x ∂μ) ≠ ⊤

case hf
α : Type u_1
β : Type ?u.432299
m : MeasurableSpace α
μ ν : Measure α
s t : SignedMeasure α
inst✝ : HaveLebesgueDecomposition s μ
⊢ (∫⁻ (x : α), Measure.rnDeriv (toJordanDecomposition s).posPart μ x ∂μ) ≠ ⊤

case hf
α : Type u_1
β : Type ?u.432299
m : MeasurableSpace α
μ ν : Measure α
s t : SignedMeasure α
inst✝ : HaveLebesgueDecomposition s μ
⊢ (∫⁻ (x : α), Measure.rnDeriv (toJordanDecomposition s).negPart μ x ∂μ) ≠ ⊤

case hf
α : Type u_1
β : Type ?u.432299
m : MeasurableSpace α
μ ν : Measure α
s t : SignedMeasure α
inst✝ : HaveLebesgueDecomposition s μ
⊢ (∫⁻ (x : α), Measure.rnDeriv (toJordanDecomposition s).posPart μ x ∂μ) ≠ ⊤

case hf
α : Type u_1
β : Type ?u.432299
m : MeasurableSpace α
μ ν : Measure α
s t : SignedMeasure α
inst✝ : HaveLebesgueDecomposition s μ
⊢ (∫⁻ (x : α), Measure.rnDeriv (toJordanDecomposition s).negPart μ x ∂μ) ≠ ⊤

case hf
α : Type u_1
β : Type ?u.432299
m : MeasurableSpace α
μ ν : Measure α
s t : SignedMeasure α
inst✝ : HaveLebesgueDecomposition s μ
⊢ (∫⁻ (x : α), Measure.rnDeriv (toJordanDecomposition s).posPart μ x ∂μ) ≠ ⊤

case hf
α : Type u_1
β : Type ?u.432299
m : MeasurableSpace α
μ ν : Measure α
s t : SignedMeasure α
inst✝ : HaveLebesgueDecomposition s μ
⊢ (∫⁻ (x : α), Measure.rnDeriv (toJordanDecomposition s).negPart μ x ∂μ) ≠ ⊤

case hf
α : Type u_1
β : Type ?u.432299
m : MeasurableSpace α
μ ν : Measure α
s t : SignedMeasure α
inst✝ : HaveLebesgueDecomposition s μ
⊢ (∫⁻ (x : α), Measure.rnDeriv (toJordanDecomposition s).posPart μ x ∂μ) ≠ ⊤

case hf
α : Type u_1
β : Type ?u.432299
m : MeasurableSpace α
μ ν : Measure α
s t : SignedMeasure α
inst✝ : HaveLebesgueDecomposition s μ
⊢ (∫⁻ (x : α), Measure.rnDeriv (toJordanDecomposition s).negPart μ x ∂μ) ≠ ⊤

case hfm
α : Type u_1
β : Type ?u.432299
m : MeasurableSpace α
μ ν : Measure α
s t : SignedMeasure α
inst✝ : HaveLebesgueDecomposition s μ
⊢ AEMeasurable fun x => Measure.rnDeriv (toJordanDecomposition s).negPart μ x

case hf
α : Type u_1
β : Type ?u.432299
m : MeasurableSpace α
μ ν : Measure α
s t : SignedMeasure α
inst✝ : HaveLebesgueDecomposition s μ
⊢ (∫⁻ (x : α), Measure.rnDeriv (toJordanDecomposition s).negPart μ x ∂μ) ≠ ⊤

case hf
α : Type u_1
β : Type ?u.432299
m : MeasurableSpace α
μ ν : Measure α
s t : SignedMeasure α
inst✝ : HaveLebesgueDecomposition s μ
⊢ (∫⁻ (x : α), Measure.rnDeriv (toJordanDecomposition s).posPart μ x ∂μ) ≠ ⊤

case hfm
α : Type u_1
β : Type ?u.432299
m : MeasurableSpace α
μ ν : Measure α
s t : SignedMeasure α
inst✝ : HaveLebesgueDecomposition s μ
⊢ AEMeasurable fun x => Measure.rnDeriv (toJordanDecomposition s).posPart μ x

case hf
α : Type u_1
β : Type ?u.432299
m : MeasurableSpace α
μ ν : Measure α
s t : SignedMeasure α
inst✝ : HaveLebesgueDecomposition s μ
⊢ (∫⁻ (x : α), Measure.rnDeriv (toJordanDecomposition s).posPart μ x ∂μ) ≠ ⊤

α : Type u_1
β : Type ?u.432299
m : MeasurableSpace α
μ ν : Measure α
s t : SignedMeasure α
inst✝ : HaveLebesgueDecomposition s μ
⊢ AEMeasurable fun x => Measure.rnDeriv (toJordanDecomposition s).posPart μ x

α : Type u_1
β : Type ?u.432299
m : MeasurableSpace α
μ ν : Measure α
s t : SignedMeasure α
inst✝ : HaveLebesgueDecomposition s μ
⊢ (∫⁻ (x : α), Measure.rnDeriv (toJordanDecomposition s).posPart μ x ∂μ) ≠ ⊤

α : Type u_1
β : Type ?u.432299
m : MeasurableSpace α
μ ν : Measure α
s t : SignedMeasure α
inst✝ : HaveLebesgueDecomposition s μ
⊢ AEMeasurable fun x => Measure.rnDeriv (toJordanDecomposition s).negPart μ x

α : Type u_1
β : Type ?u.432299
m : MeasurableSpace α
μ ν : Measure α
s t : SignedMeasure α
inst✝ : HaveLebesgueDecomposition s μ
⊢ (∫⁻ (x : α), Measure.rnDeriv (toJordanDecomposition s).negPart μ x ∂μ) ≠ ⊤

tactic: all_goals
  first
  | exact (lintegral_rnDeriv_lt_top _ _).ne
  | measurability
state_before: case hf
α : Type u_1
β : Type ?u.432299
m : MeasurableSpace α
μ ν : Measure α
s t : SignedMeasure α
inst✝ : HaveLebesgueDecomposition s μ
⊢ (∫⁻ (x : α), Measure.rnDeriv (toJordanDecomposition s).posPart μ x ∂μ) ≠ ⊤

case hf
α : Type u_1
β : Type ?u.432299
m : MeasurableSpace α
μ ν : Measure α
s t : SignedMeasure α
inst✝ : HaveLebesgueDecomposition s μ
⊢ (∫⁻ (x : α), Measure.rnDeriv (toJordanDecomposition s).negPart μ x ∂μ) ≠ ⊤

case hf
α : Type u_1
β : Type ?u.432299
m : MeasurableSpace α
μ ν : Measure α
s t : SignedMeasure α
inst✝ : HaveLebesgueDecomposition s μ
⊢ (∫⁻ (x : α), Measure.rnDeriv (toJordanDecomposition s).negPart μ x ∂μ) ≠ ⊤

case hf
α : Type u_1
β : Type ?u.432299
m : MeasurableSpace α
μ ν : Measure α
s t : SignedMeasure α
inst✝ : HaveLebesgueDecomposition s μ
⊢ (∫⁻ (x : α), Measure.rnDeriv (toJordanDecomposition s).posPart μ x ∂μ) ≠ ⊤

case hf
α : Type u_1
β : Type ?u.432299
m : MeasurableSpace α
μ ν : Measure α
s t : SignedMeasure α
inst✝ : HaveLebesgueDecomposition s μ
⊢ (∫⁻ (x : α), Measure.rnDeriv (toJordanDecomposition s).negPart μ x ∂μ) ≠ ⊤

case hf
α : Type u_1
β : Type ?u.432299
m : MeasurableSpace α
μ ν : Measure α
s t : SignedMeasure α
inst✝ : HaveLebesgueDecomposition s μ
⊢ (∫⁻ (x : α), Measure.rnDeriv (toJordanDecomposition s).posPart μ x ∂μ) ≠ ⊤

case hf
α : Type u_1
β : Type ?u.432299
m : MeasurableSpace α
μ ν : Measure α
s t : SignedMeasure α
inst✝ : HaveLebesgueDecomposition s μ
⊢ (∫⁻ (x : α), Measure.rnDeriv (toJordanDecomposition s).negPart μ x ∂μ) ≠ ⊤

case hf
α : Type u_1
β : Type ?u.432299
m : MeasurableSpace α
μ ν : Measure α
s t : SignedMeasure α
inst✝ : HaveLebesgueDecomposition s μ
⊢ (∫⁻ (x : α), Measure.rnDeriv (toJordanDecomposition s).posPart μ x ∂μ) ≠ ⊤

case hf
α : Type u_1
β : Type ?u.432299
m : MeasurableSpace α
μ ν : Measure α
s t : SignedMeasure α
inst✝ : HaveLebesgueDecomposition s μ
⊢ (∫⁻ (x : α), Measure.rnDeriv (toJordanDecomposition s).negPart μ x ∂μ) ≠ ⊤

case hf
α : Type u_1
β : Type ?u.432299
m : MeasurableSpace α
μ ν : Measure α
s t : SignedMeasure α
inst✝ : HaveLebesgueDecomposition s μ
⊢ (∫⁻ (x : α), Measure.rnDeriv (toJordanDecomposition s).posPart μ x ∂μ) ≠ ⊤

case hf
α : Type u_1
β : Type ?u.432299
m : MeasurableSpace α
μ ν : Measure α
s t : SignedMeasure α
inst✝ : HaveLebesgueDecomposition s μ
⊢ (∫⁻ (x : α), Measure.rnDeriv (toJordanDecomposition s).posPart μ x ∂μ) ≠ ⊤

case hf
α : Type u_1
β : Type ?u.432299
m : MeasurableSpace α
μ ν : Measure α
s t : SignedMeasure α
inst✝ : HaveLebesgueDecomposition s μ
⊢ (∫⁻ (x : α), Measure.rnDeriv (toJordanDecomposition s).negPart μ x ∂μ) ≠ ⊤

case hf
α : Type u_1
β : Type ?u.432299
m : MeasurableSpace α
μ ν : Measure α
s t : SignedMeasure α
inst✝ : HaveLebesgueDecomposition s μ
⊢ (∫⁻ (x : α), Measure.rnDeriv (toJordanDecomposition s).posPart μ x ∂μ) ≠ ⊤

case hf
α : Type u_1
β : Type ?u.432299
m : MeasurableSpace α
μ ν : Measure α
s t : SignedMeasure α
inst✝ : HaveLebesgueDecomposition s μ
⊢ (∫⁻ (x : α), Measure.rnDeriv (toJordanDecomposition s).negPart μ x ∂μ) ≠ ⊤

case hf
α : Type u_1
β : Type ?u.432299
m : MeasurableSpace α
μ ν : Measure α
s t : SignedMeasure α
inst✝ : HaveLebesgueDecomposition s μ
⊢ (∫⁻ (x : α), Measure.rnDeriv (toJordanDecomposition s).posPart μ x ∂μ) ≠ ⊤

case hf
α : Type u_1
β : Type ?u.432299
m : MeasurableSpace α
μ ν : Measure α
s t : SignedMeasure α
inst✝ : HaveLebesgueDecomposition s μ
⊢ (∫⁻ (x : α), Measure.rnDeriv (toJordanDecomposition s).negPart μ x ∂μ) ≠ ⊤

case hf
α : Type u_1
β : Type ?u.432299
m : MeasurableSpace α
μ ν : Measure α
s t : SignedMeasure α
inst✝ : HaveLebesgueDecomposition s μ
⊢ (∫⁻ (x : α), Measure.rnDeriv (toJordanDecomposition s).posPart μ x ∂μ) ≠ ⊤

case hf
α : Type u_1
β : Type ?u.432299
m : MeasurableSpace α
μ ν : Measure α
s t : SignedMeasure α
inst✝ : HaveLebesgueDecomposition s μ
⊢ (∫⁻ (x : α), Measure.rnDeriv (toJordanDecomposition s).negPart μ x ∂μ) ≠ ⊤

case hf
α : Type u_1
β : Type ?u.432299
m : MeasurableSpace α
μ ν : Measure α
s t : SignedMeasure α
inst✝ : HaveLebesgueDecomposition s μ
⊢ (∫⁻ (x : α), Measure.rnDeriv (toJordanDecomposition s).posPart μ x ∂μ) ≠ ⊤

case hf
α : Type u_1
β : Type ?u.432299
m : MeasurableSpace α
μ ν : Measure α
s t : SignedMeasure α
inst✝ : HaveLebesgueDecomposition s μ
⊢ (∫⁻ (x : α), Measure.rnDeriv (toJordanDecomposition s).negPart μ x ∂μ) ≠ ⊤

case hf
α : Type u_1
β : Type ?u.432299
m : MeasurableSpace α
μ ν : Measure α
s t : SignedMeasure α
inst✝ : HaveLebesgueDecomposition s μ
⊢ (∫⁻ (x : α), Measure.rnDeriv (toJordanDecomposition s).posPart μ x ∂μ) ≠ ⊤

case hf
α : Type u_1
β : Type ?u.432299
m : MeasurableSpace α
μ ν : Measure α
s t : SignedMeasure α
inst✝ : HaveLebesgueDecomposition s μ
⊢ (∫⁻ (x : α), Measure.rnDeriv (toJordanDecomposition s).negPart μ x ∂μ) ≠ ⊤

case hf
α : Type u_1
β : Type ?u.432299
m : MeasurableSpace α
μ ν : Measure α
s t : SignedMeasure α
inst✝ : HaveLebesgueDecomposition s μ
⊢ (∫⁻ (x : α), Measure.rnDeriv (toJordanDecomposition s).posPart μ x ∂μ) ≠ ⊤

case hf
α : Type u_1
β : Type ?u.432299
m : MeasurableSpace α
μ ν : Measure α
s t : SignedMeasure α
inst✝ : HaveLebesgueDecomposition s μ
⊢ (∫⁻ (x : α), Measure.rnDeriv (toJordanDecomposition s).negPart μ x ∂μ) ≠ ⊤

case hfm
α : Type u_1
β : Type ?u.432299
m : MeasurableSpace α
μ ν : Measure α
s t : SignedMeasure α
inst✝ : HaveLebesgueDecomposition s μ
⊢ AEMeasurable fun x => Measure.rnDeriv (toJordanDecomposition s).negPart μ x

case hf
α : Type u_1
β : Type ?u.432299
m : MeasurableSpace α
μ ν : Measure α
s t : SignedMeasure α
inst✝ : HaveLebesgueDecomposition s μ
⊢ (∫⁻ (x : α), Measure.rnDeriv (toJordanDecomposition s).negPart μ x ∂μ) ≠ ⊤

case hf
α : Type u_1
β : Type ?u.432299
m : MeasurableSpace α
μ ν : Measure α
s t : SignedMeasure α
inst✝ : HaveLebesgueDecomposition s μ
⊢ (∫⁻ (x : α), Measure.rnDeriv (toJordanDecomposition s).posPart μ x ∂μ) ≠ ⊤

case hfm
α : Type u_1
β : Type ?u.432299
m : MeasurableSpace α
μ ν : Measure α
s t : SignedMeasure α
inst✝ : HaveLebesgueDecomposition s μ
⊢ AEMeasurable fun x => Measure.rnDeriv (toJordanDecomposition s).posPart μ x

case hf
α : Type u_1
β : Type ?u.432299
m : MeasurableSpace α
μ ν : Measure α
s t : SignedMeasure α
inst✝ : HaveLebesgueDecomposition s μ
⊢ (∫⁻ (x : α), Measure.rnDeriv (toJordanDecomposition s).posPart μ x ∂μ) ≠ ⊤

α : Type u_1
β : Type ?u.432299
m : MeasurableSpace α
μ ν : Measure α
s t : SignedMeasure α
inst✝ : HaveLebesgueDecomposition s μ
⊢ AEMeasurable fun x => Measure.rnDeriv (toJordanDecomposition s).posPart μ x

α : Type u_1
β : Type ?u.432299
m : MeasurableSpace α
μ ν : Measure α
s t : SignedMeasure α
inst✝ : HaveLebesgueDecomposition s μ
⊢ (∫⁻ (x : α), Measure.rnDeriv (toJordanDecomposition s).posPart μ x ∂μ) ≠ ⊤

α : Type u_1
β : Type ?u.432299
m : MeasurableSpace α
μ ν : Measure α
s t : SignedMeasure α
inst✝ : HaveLebesgueDecomposition s μ
⊢ AEMeasurable fun x => Measure.rnDeriv (toJordanDecomposition s).negPart μ x

α : Type u_1
β : Type ?u.432299
m : MeasurableSpace α
μ ν : Measure α
s t : SignedMeasure α
inst✝ : HaveLebesgueDecomposition s μ
⊢ (∫⁻ (x : α), Measure.rnDeriv (toJordanDecomposition s).negPart μ x ∂μ) ≠ ⊤

tactic: exact s.toJordanDecomposition.posPart.haveLebesgueDecomposition_add μ
state_before: case h.e'_3.h.e'_5.h.e'_3
α : Type u_1
β : Type ?u.432299
m : MeasurableSpace α
μ ν : Measure α
s t : SignedMeasure α
inst✝ : HaveLebesgueDecomposition s μ
⊢ (toJordanDecomposition s).posPart =
    Measure.singularPart (toJordanDecomposition s).posPart μ +
      withDensity μ fun x => Measure.rnDeriv (toJordanDecomposition s).posPart μ x

tactic: rw [add_comm]
state_before: case h.e'_3.h.e'_6.h.e'_3
α : Type u_1
β : Type ?u.432299
m : MeasurableSpace α
μ ν : Measure α
s t : SignedMeasure α
inst✝ : HaveLebesgueDecomposition s μ
⊢ (toJordanDecomposition s).negPart =
    (withDensity μ fun x => Measure.rnDeriv (toJordanDecomposition s).negPart μ x) +
      Measure.singularPart (toJordanDecomposition s).negPart μ

tactic: exact s.toJordanDecomposition.negPart.haveLebesgueDecomposition_add μ
state_before: case h.e'_3.h.e'_6.h.e'_3
α : Type u_1
β : Type ?u.432299
m : MeasurableSpace α
μ ν : Measure α
s t : SignedMeasure α
inst✝ : HaveLebesgueDecomposition s μ
⊢ (toJordanDecomposition s).negPart =
    Measure.singularPart (toJordanDecomposition s).negPart μ +
      withDensity μ fun x => Measure.rnDeriv (toJordanDecomposition s).negPart μ x

tactic: first
| exact (lintegral_rnDeriv_lt_top _ _).ne
| measurability
state_before: α : Type u_1
β : Type ?u.432299
m : MeasurableSpace α
μ ν : Measure α
s t : SignedMeasure α
inst✝ : HaveLebesgueDecomposition s μ
⊢ (∫⁻ (x : α), Measure.rnDeriv (toJordanDecomposition s).negPart μ x ∂μ) ≠ ⊤

tactic: exact (lintegral_rnDeriv_lt_top _ _).ne
state_before: α : Type u_1
β : Type ?u.432299
m : MeasurableSpace α
μ ν : Measure α
s t : SignedMeasure α
inst✝ : HaveLebesgueDecomposition s μ
⊢ (∫⁻ (x : α), Measure.rnDeriv (toJordanDecomposition s).negPart μ x ∂μ) ≠ ⊤

tactic: measurability
state_before: α : Type u_1
β : Type ?u.432299
m : MeasurableSpace α
μ ν : Measure α
s t : SignedMeasure α
inst✝ : HaveLebesgueDecomposition s μ
⊢ AEMeasurable fun x => Measure.rnDeriv (toJordanDecomposition s).negPart μ x