Theorems · Theorem · measure theory

MeasureTheory.Measure.measure_prod_null

∀ {α : Type u_1} {β : Type u_2} [inst : MeasurableSpace α] [inst_1 : MeasurableSpace β] {μ : MeasureTheory.Measure α}
  {ν : MeasureTheory.Measure β} [MeasureTheory.SFinite ν] {s : Set (α × β)},
  MeasurableSet s → ((μ.prod ν) s = 0 ↔ (fun x => ν (Prod.mk x ⁻¹' s)) =ᵐ[μ] 0)

A measurable set s has μ.prod ν measure zero, where ν is an s-finite measure, if and only if μ-a.e. section {y | (x, y) ∈ s} of s have ν measure zero. See measure_ae_null_of_prod_null for the forward implication without the measurability assumption and measure_prod_null_of_ae_null for the reverse implication without the s-finiteness assumption. Note: the assumption hs cannot be dropped. For a counterexample, see Walter Rudin Real and Complex Analysis, example (c) in section 8.9.

Defined in

Mathlib.MeasureTheory.Measure.Prod

Cited by

4 results in Mathlib

Foundations

Depth 216 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

MeasurableSpaceMeasurableSpaceMeasureTheory.SFinite