File: Mathlib/MeasureTheory/Measure/Complex.lean

full_name: MeasureTheory.ComplexMeasure.re
code: /-- The real part of a complex measure is a signed measure. -/
@[simps! apply]
def re : ComplexMeasure α →ₗ[ℝ] SignedMeasure α 
goal: MeasureTheory.ComplexMeasure α →ₗ[ℝ] MeasureTheory.SignedMeasure α

full_name: MeasureTheory.ComplexMeasure.re_apply
code: /-- The real part of a complex measure is a signed measure. -/
@[simps! apply]
def re : ComplexMeasure α →ₗ[ℝ] SignedMeasure α 
goal: MeasureTheory.ComplexMeasure.re v = v.mapRange Complex.reLm.toAddMonoidHom Complex.continuous_re

full_name: MeasureTheory.ComplexMeasure.im
code: /-- The imaginary part of a complex measure is a signed measure. -/
@[simps! apply]
def im : ComplexMeasure α →ₗ[ℝ] SignedMeasure α 
goal: MeasureTheory.ComplexMeasure α →ₗ[ℝ] MeasureTheory.SignedMeasure α

full_name: MeasureTheory.ComplexMeasure.im_apply
code: /-- The imaginary part of a complex measure is a signed measure. -/
@[simps! apply]
def im : ComplexMeasure α →ₗ[ℝ] SignedMeasure α 
goal: MeasureTheory.ComplexMeasure.im v = v.mapRange Complex.imLm.toAddMonoidHom Complex.continuous_im

full_name: MeasureTheory.SignedMeasure.toComplexMeasure_apply_im
code: /-- Given `s` and `t` signed measures, `s + it` is a complex measure-/
@[simps!]
def _root_.MeasureTheory.SignedMeasure.toComplexMeasure (s t : SignedMeasure α) :
    ComplexMeasure α 
goal: (↑(s.toComplexMeasure t) i).im = ↑t i

full_name: MeasureTheory.SignedMeasure.toComplexMeasure_apply_re
code: /-- Given `s` and `t` signed measures, `s + it` is a complex measure-/
@[simps!]
def _root_.MeasureTheory.SignedMeasure.toComplexMeasure (s t : SignedMeasure α) :
    ComplexMeasure α 
goal: (↑(s.toComplexMeasure t) i).re = ↑s i

full_name: MeasureTheory.SignedMeasure.toComplexMeasure
code: /-- Given `s` and `t` signed measures, `s + it` is a complex measure-/
@[simps!]
def _root_.MeasureTheory.SignedMeasure.toComplexMeasure (s t : SignedMeasure α) :
    ComplexMeasure α 
goal: MeasureTheory.ComplexMeasure α

full_name: MeasureTheory.SignedMeasure.toComplexMeasure_apply
code: theorem _root_.MeasureTheory.SignedMeasure.toComplexMeasure_apply
    {s t : SignedMeasure α} {i : Set α} : s.toComplexMeasure t i = ⟨s i, t i⟩ 
goal: ↑(s.toComplexMeasure t) i = { re := ↑s i, im := ↑t i }

full_name: MeasureTheory.ComplexMeasure.toComplexMeasure_to_signedMeasure
code: theorem toComplexMeasure_to_signedMeasure (c : ComplexMeasure α) :
    SignedMeasure.toComplexMeasure (ComplexMeasure.re c) (ComplexMeasure.im c) = c 
goal: (MeasureTheory.ComplexMeasure.re c).toComplexMeasure (MeasureTheory.ComplexMeasure.im c) = c

full_name: MeasureTheory.SignedMeasure.re_toComplexMeasure
code: theorem _root_.MeasureTheory.SignedMeasure.re_toComplexMeasure (s t : SignedMeasure α) :
    ComplexMeasure.re (SignedMeasure.toComplexMeasure s t) = s 
goal: MeasureTheory.ComplexMeasure.re (s.toComplexMeasure t) = s

full_name: MeasureTheory.SignedMeasure.im_toComplexMeasure
code: theorem _root_.MeasureTheory.SignedMeasure.im_toComplexMeasure (s t : SignedMeasure α) :
    ComplexMeasure.im (SignedMeasure.toComplexMeasure s t) = t 
goal: MeasureTheory.ComplexMeasure.im (s.toComplexMeasure t) = t

full_name: MeasureTheory.ComplexMeasure.equivSignedMeasure
code: /-- The complex measures form an equivalence to the type of pairs of signed measures. -/
@[simps]
def equivSignedMeasure : ComplexMeasure α ≃ SignedMeasure α × SignedMeasure α 
goal: MeasureTheory.ComplexMeasure α ≃ MeasureTheory.SignedMeasure α × MeasureTheory.SignedMeasure α

full_name: MeasureTheory.ComplexMeasure.equivSignedMeasure_apply
code: /-- The complex measures form an equivalence to the type of pairs of signed measures. -/
@[simps]
def equivSignedMeasure : ComplexMeasure α ≃ SignedMeasure α × SignedMeasure α 
goal: MeasureTheory.ComplexMeasure.equivSignedMeasure c =
  (MeasureTheory.ComplexMeasure.re c, MeasureTheory.ComplexMeasure.im c)

full_name: MeasureTheory.ComplexMeasure.equivSignedMeasure_symm_apply
code: /-- The complex measures form an equivalence to the type of pairs of signed measures. -/
@[simps]
def equivSignedMeasure : ComplexMeasure α ≃ SignedMeasure α × SignedMeasure α 
goal: ∀ (x : MeasureTheory.SignedMeasure α × MeasureTheory.SignedMeasure α),
  MeasureTheory.ComplexMeasure.equivSignedMeasure.symm x =
    match x with
    | (s, t) => s.toComplexMeasure t

full_name: MeasureTheory.ComplexMeasure.equivSignedMeasureₗ_symm_apply
code: /-- The complex measures form a linear isomorphism to the type of pairs of signed measures. -/
@[simps]
def equivSignedMeasureₗ : ComplexMeasure α ≃ₗ[R] SignedMeasure α × SignedMeasure α 
goal: ∀ (a : MeasureTheory.SignedMeasure α × MeasureTheory.SignedMeasure α),
  MeasureTheory.ComplexMeasure.equivSignedMeasureₗ.symm a = MeasureTheory.ComplexMeasure.equivSignedMeasure.invFun a

full_name: MeasureTheory.ComplexMeasure.equivSignedMeasureₗ_apply
code: /-- The complex measures form a linear isomorphism to the type of pairs of signed measures. -/
@[simps]
def equivSignedMeasureₗ : ComplexMeasure α ≃ₗ[R] SignedMeasure α × SignedMeasure α 
goal: ∀ (a : MeasureTheory.ComplexMeasure α),
  MeasureTheory.ComplexMeasure.equivSignedMeasureₗ a = MeasureTheory.ComplexMeasure.equivSignedMeasure.toFun a

full_name: MeasureTheory.ComplexMeasure.equivSignedMeasureₗ
code: /-- The complex measures form a linear isomorphism to the type of pairs of signed measures. -/
@[simps]
def equivSignedMeasureₗ : ComplexMeasure α ≃ₗ[R] SignedMeasure α × SignedMeasure α 
goal: MeasureTheory.ComplexMeasure α ≃ₗ[R] MeasureTheory.SignedMeasure α × MeasureTheory.SignedMeasure α

full_name: MeasureTheory.ComplexMeasure.absolutelyContinuous_ennreal_iff
code: theorem absolutelyContinuous_ennreal_iff (c : ComplexMeasure α) (μ : VectorMeasure α ℝ≥0∞) :
    c ≪ᵥ μ ↔ ComplexMeasure.re c ≪ᵥ μ ∧ ComplexMeasure.im c ≪ᵥ μ 
goal: c ≪ᵥ μ ↔ MeasureTheory.ComplexMeasure.re c ≪ᵥ μ ∧ MeasureTheory.ComplexMeasure.im c ≪ᵥ μ