Lebesgue measure on ℂ

In this file, we consider the Lebesgue measure on ℂ defined as the push-forward of the volume on ℝ² under the natural isomorphism and prove that it is equal to the measure volume of ℂ coming from its InnerProductSpace structure over ℝ. For that, we consider the two frequently used ways to represent ℝ² in mathlib: ℝ × ℝ and Fin 2 → ℝ, define measurable equivalences (MeasurableEquiv) to both types and prove that both of them are volume preserving (in the sense of MeasureTheory.measurePreserving).

def Complex.measurableEquivPi : ℂ ≃ᵐ (Fin 2 → ℝ)

Measurable equivalence between ℂ and ℝ² = Fin 2 → ℝ.

Equations

• Complex.measurableEquivPi = Homeomorph.toMeasurableEquiv (ContinuousLinearEquiv.toHomeomorph (LinearEquiv.toContinuousLinearEquiv (Basis.equivFun Complex.basisOneI)))

def Complex.measurableEquivRealProd : ℂ ≃ᵐ ℝ × ℝ

Measurable equivalence between ℂ and ℝ × ℝ.

Equations

• Complex.measurableEquivRealProd = Homeomorph.toMeasurableEquiv (ContinuousLinearEquiv.toHomeomorph Complex.equivRealProdClm)

theorem Complex.volume_preserving_equiv_pi : MeasureTheory.MeasurePreserving ↑Complex.measurableEquivPi

theorem Complex.volume_preserving_equiv_real_prod : MeasureTheory.MeasurePreserving ↑Complex.measurableEquivRealProd