Theorems · Theorem · measure theory

Measure.ext_of_lintegral_prod_mul_prod_boundedContinuousFunction

∀ {ι : Type u_1} {κ : Type u_2} {X : ι → Type u_5} {Y : κ → Type u_6} {mX : (i : ι) → MeasurableSpace (X i)}
  [inst : (i : ι) → TopologicalSpace (X i)] [∀ (i : ι), BorelSpace (X i)] [∀ (i : ι), HasOuterApproxClosed (X i)]
  {mY : (j : κ) → MeasurableSpace (Y j)} [inst_3 : (j : κ) → TopologicalSpace (Y j)] [∀ (j : κ), BorelSpace (Y j)]
  [∀ (j : κ), HasOuterApproxClosed (Y j)] [inst_6 : Fintype ι] [inst_7 : Fintype κ]
  {μ ν : MeasureTheory.Measure (((i : ι) → X i) × ((j : κ) → Y j))} [MeasureTheory.IsFiniteMeasure μ],
  (∀ (f : (i : ι) → BoundedContinuousFunction (X i) NNReal) (g : (j : κ) → BoundedContinuousFunction (Y j) NNReal),
      ∫⁻ (p : ((i : ι) → X i) × ((j : κ) → Y j)), ↑(∏ i, (f i) (p.1 i)) * ↑(∏ j, (g j) (p.2 j)) ∂μ =
        ∫⁻ (p : ((i : ι) → X i) × ((j : κ) → Y j)), ↑(∏ i, (f i) (p.1 i)) * ↑(∏ j, (g j) (p.2 j)) ∂ν) →
    μ = ν

A finite measure μ over (Π i, X i) × (Π j, Y j) is determined by the values ∫⁻ p, (Π i, f i (p.1 i)) * (Π j, g j (p.2 j)) ∂μ, for f : (i : ι) → X i → ℝ≥0 and g : (j : κ) → Y j → ℝ≥0 any families of bounded continuous functions.

Defined in

Mathlib.MeasureTheory.Measure.HasOuterApproxClosedProd

Cited by

1 results in Mathlib

Foundations

Depth 215 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

TopologicalSpaceBorelSpaceHasOuterApproxClosedTopologicalSpaceBorelSpaceHasOuterApproxClosedFintypeFintypeMeasureTheory.IsFiniteMeasure