Theorems · Theorem · measure theory

MeasureTheory.Measure.measure_prod_null_of_ae_null

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

If μ-a.e. section {y | (x, y) ∈ s} of a measurable set have ν measure zero, then s has μ.prod ν measure zero. This implication requires s to be measurable but does not require ν to be s-finite. See also measure_prod_null and measure_ae_null_of_prod_null below.

Defined in

Mathlib.MeasureTheory.Measure.Prod

Cited by

1 results in Mathlib

Foundations

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

Assumes

MeasurableSpaceMeasurableSpace