Theorems · Theorem · measure theory

measurable_measure_prodMk_left

∀ {α : Type u_1} {β : Type u_2} [inst : MeasurableSpace α] [inst_1 : MeasurableSpace β] {ν : MeasureTheory.Measure β}
  [MeasureTheory.SFinite ν] {s : Set (α × β)}, MeasurableSet s → Measurable fun x => ν (Prod.mk x ⁻¹' s)

If ν is an s-finite measure, and s ⊆ α × β is measurable, then x ↦ ν { y | (x, y) ∈ s } is a measurable function. Not true without the s-finite assumption: on ℝ × ℝ with the product sigma-algebra, let s be the diagonal and let ν be an uncountable sum of Dirac measures (all Dirac measures for points in a set t). Then ν (Prod.mk x ⁻¹' s) = ν {x} = if x ∈ t then 1 else 0. If t is chosen non-measurable, this will not be measurable.

Defined in

Mathlib.MeasureTheory.Measure.Prod

Cited by

20 results in Mathlib

Foundations

Depth 195 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

MeasurableSpaceMeasurableSpaceMeasureTheory.SFinite