Theorems · Theorem · measure theory

MeasureTheory.Measure.map_eq_sum

∀ {α : Type u_1} {β : Type u_2} [inst : MeasurableSpace α] [inst_1 : MeasurableSpace β] [Countable β]
  [MeasurableSingletonClass β] (μ : MeasureTheory.Measure α) (f : α → β),
  Measurable f →
    MeasureTheory.Measure.map f μ = MeasureTheory.Measure.sum fun b => μ (f ⁻¹' {b}) • MeasureTheory.Measure.dirac b

If f is a map with countable codomain, then μ.map f is a sum of Dirac measures.

Defined in

Mathlib.MeasureTheory.Measure.Dirac

Cited by

1 results in Mathlib

Foundations

Depth 200 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

MeasurableSpaceMeasurableSpaceCountableMeasurableSingletonClass