Theorems · Theorem · measure theory

MeasureTheory.Measure.prod_eq

∀ {α : Type u_1} {β : Type u_2} [inst : MeasurableSpace α] [inst_1 : MeasurableSpace β] {μ : MeasureTheory.Measure α}
  [MeasureTheory.SigmaFinite μ] {ν : MeasureTheory.Measure β} [MeasureTheory.SigmaFinite ν]
  {μν : MeasureTheory.Measure (α × β)},
  (∀ (s : Set α) (t : Set β), MeasurableSet s → MeasurableSet t → μν (s ×ˢ t) = μ s * ν t) → μ.prod ν = μν

A measure on a product space equals the product measure of sigma-finite measures if they are equal on rectangles.

Defined in

Mathlib.MeasureTheory.Measure.Prod

Cited by

12 results in Mathlib

Foundations

Depth 221 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

MeasurableSpaceMeasurableSpaceMeasureTheory.SigmaFiniteMeasureTheory.SigmaFinite