Theorems · Theorem · measure theory

MeasureTheory.Measure.prod_eq_generateFrom

∀ {α : Type u_1} {β : Type u_2} [inst : MeasurableSpace α] [inst_1 : MeasurableSpace β] {μ : MeasureTheory.Measure α}
  {ν : MeasureTheory.Measure β} {C : Set (Set α)} {D : Set (Set β)},
  MeasurableSpace.generateFrom C = inst →
    MeasurableSpace.generateFrom D = inst_1 →
      IsPiSystem C →
        IsPiSystem D →
          ∀ (h3C : μ.FiniteSpanningSetsIn C) (h3D : ν.FiniteSpanningSetsIn D) {μν : MeasureTheory.Measure (α × β)},
            (∀ s ∈ C, ∀ t ∈ D, μν (s ×ˢ t) = μ s * ν t) → μ.prod ν = μν

A measure on a product space equals the product measure if they are equal on rectangles with as sides sets that generate the corresponding σ-algebras.

Defined in

Mathlib.MeasureTheory.Measure.Prod

Cited by

3 results in Mathlib

Foundations

Depth 220 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

MeasurableSpaceMeasurableSpace