Theorem: _private.Mathlib.MeasureTheory.Decomposition.SignedLebesgue.0.MeasureTheory.SignedMeasure.haveLebesgueDecomposition_mk'
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 : SignedMeasure α
μ : Measure α
f : α → ℝ
hf : Measurable f
hfi : Integrable f μ
htμ : t ⟂ᵥ μ.toENNRealVectorMeasure
hadd : s = t + μ.withDensityᵥ f
⊢ s.HaveLebesgueDecomposition μ
state_after: α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ✝ ν : Measure α
s t : SignedMeasure α
μ : Measure α
f : α → ℝ
hf : Measurable f
hfi : Integrable f μ
htμ : t ⟂ᵥ μ.toENNRealVectorMeasure
hadd : s = t + μ.withDensityᵥ f
htμ' : t ⟂ᵥ μ.toENNRealVectorMeasure
⊢ s.HaveLebesgueDecomposition μ
tactic: have htμ' := htμ
state_before: α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ✝ ν : Measure α
s t : SignedMeasure α
μ : Measure α
f : α → ℝ
hf : Measurable f
hfi : Integrable f μ
htμ : t ⟂ᵥ μ.toENNRealVectorMeasure
hadd : s = t + μ.withDensityᵥ f
htμ' : t ⟂ᵥ μ.toENNRealVectorMeasure
⊢ s.HaveLebesgueDecomposition μ
state_after: α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ✝ ν : Measure α
s t : SignedMeasure α
μ : Measure α
f : α → ℝ
hf : Measurable f
hfi : Integrable f μ
htμ : t.totalVariation ⟂ₘ μ.toENNRealVectorMeasure.ennrealToMeasure
hadd : s = t + μ.withDensityᵥ f
htμ' : t ⟂ᵥ μ.toENNRealVectorMeasure
⊢ s.HaveLebesgueDecomposition μ
tactic: rw [mutuallySingular_ennreal_iff] at htμ
state_before: α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ✝ ν : Measure α
s t : SignedMeasure α
μ : Measure α
f : α → ℝ
hf : Measurable f
hfi : Integrable f μ
htμ : t.totalVariation ⟂ₘ μ.toENNRealVectorMeasure.ennrealToMeasure
hadd : s = t + μ.withDensityᵥ f
htμ' : t ⟂ᵥ μ.toENNRealVectorMeasure
⊢ s.HaveLebesgueDecomposition μ
state_after: α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ✝ ν : Measure α
s t : SignedMeasure α
μ : Measure α
f : α → ℝ
hf : Measurable f
hfi : Integrable f μ
hadd : s = t + μ.withDensityᵥ f
htμ' : t ⟂ᵥ μ.toENNRealVectorMeasure
htμ : t.totalVariation ⟂ₘ VectorMeasure.equivMeasure.toFun (VectorMeasure.equivMeasure.invFun μ)
⊢ s.HaveLebesgueDecomposition μ
tactic: change _ ⟂ₘ VectorMeasure.equivMeasure.toFun (VectorMeasure.equivMeasure.invFun μ) at htμ
state_before: α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ✝ ν : Measure α
s t : SignedMeasure α
μ : Measure α
f : α → ℝ
hf : Measurable f
hfi : Integrable f μ
hadd : s = t + μ.withDensityᵥ f
htμ' : t ⟂ᵥ μ.toENNRealVectorMeasure
htμ : t.totalVariation ⟂ₘ VectorMeasure.equivMeasure.toFun (VectorMeasure.equivMeasure.invFun μ)
⊢ s.HaveLebesgueDecomposition μ
state_after: α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ✝ ν : Measure α
s t : SignedMeasure α
μ : Measure α
f : α → ℝ
hf : Measurable f
hfi : Integrable f μ
hadd : s = t + μ.withDensityᵥ f
htμ' : t ⟂ᵥ μ.toENNRealVectorMeasure
htμ : t.toJordanDecomposition.posPart ⟂ₘ μ ∧ t.toJordanDecomposition.negPart ⟂ₘ μ
⊢ s.HaveLebesgueDecomposition μ
tactic: rw [VectorMeasure.equivMeasure.right_inv, totalVariation_mutuallySingular_iff] at htμ
state_before: α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ✝ ν : Measure α
s t : SignedMeasure α
μ : Measure α
f : α → ℝ
hf : Measurable f
hfi : Integrable f μ
hadd : s = t + μ.withDensityᵥ f
htμ' : t ⟂ᵥ μ.toENNRealVectorMeasure
htμ : t.toJordanDecomposition.posPart ⟂ₘ μ ∧ t.toJordanDecomposition.negPart ⟂ₘ μ
⊢ s.HaveLebesgueDecomposition μ
state_after: no goals
tactic: refine
    { posPart := by
        use ⟨t.toJordanDecomposition.posPart, fun x => ENNReal.ofReal (f x)⟩
        refine ⟨hf.ennreal_ofReal, htμ.1, ?_⟩
        rw [toJordanDecomposition_eq_of_eq_add_withDensity hf hfi htμ' hadd]
      negPart := by
        use ⟨t.toJordanDecomposition.negPart, fun x => ENNReal.ofReal (-f x)⟩
        refine ⟨hf.neg.ennreal_ofReal, htμ.2, ?_⟩
        rw [toJordanDecomposition_eq_of_eq_add_withDensity hf hfi htμ' hadd] }