Theorems · Theorem · measure theory

MeasurableEmbedding.lintegral_map

∀ {α : Type u_1} {β : Type u_2} [inst : MeasurableSpace α] [inst_1 : MeasurableSpace β] {μ : MeasureTheory.Measure α}
  {g : α → β},
  MeasurableEmbedding g → ∀ (f : β → ENNReal), ∫⁻ (a : β), f a ∂MeasureTheory.Measure.map g μ = ∫⁻ (a : α), f (g a) ∂μ

If g : α → β is a measurable embedding and f : β → ℝ≥0∞ is any function (not necessarily measurable), then ∫⁻ a, f a ∂(map g μ) = ∫⁻ a, f (g a) ∂μ. Compare with lintegral_map which applies to any measurable g : α → β but requires that f is measurable as well.

Defined in

Mathlib.MeasureTheory.Integral.Lebesgue.Map

Cited by

10 results in Mathlib

Foundations

Depth 202 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

MeasurableSpaceMeasurableSpace