Theorems · Theorem · measure theory

MeasureTheory.Measure.measure_ae_null_of_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)},
  (μ.prod ν) s = 0 → (fun x => ν (Prod.mk x ⁻¹' s)) =ᵐ[μ] 0

Note: the converse is not true without assuming that s is measurable. For a counterexample, see Walter Rudin Real and Complex Analysis, example (c) in section 8.9.

Defined in

Mathlib.MeasureTheory.Measure.Prod

Cited by

1 results in Mathlib

Foundations

Depth 217 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

MeasurableSpaceMeasurableSpaceMeasureTheory.SFinite