Theorems · Theorem · measure theory

MeasureTheory.Measure.ae_ae_of_ae_prod

∀ {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] {p : α × β → Prop},
  (∀ᵐ (z : α × β) ∂μ.prod ν, p z) → ∀ᵐ (x : α) ∂μ, ∀ᵐ (y : β) ∂ν, p (x, y)

Note: the converse is not true. For a counterexample, see Walter Rudin Real and Complex Analysis, example (c) in section 8.9. It is true if the set is measurable, see ae_prod_mem_iff_ae_ae_mem.

Defined in

Mathlib.MeasureTheory.Measure.Prod

Cited by

7 results in Mathlib

Foundations

Depth 218 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

MeasurableSpaceMeasurableSpaceMeasureTheory.SFinite