Lebesgue measure on #

In this file, we consider the Lebesgue measure on 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 coming from its InnerProductSpace structure over . 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).

theorem Complex.measurableEquivPi_symm_apply (p : Fin 2) :
Complex.measurableEquivPi.symm p = (p 0) + (p 1) * Complex.I