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For sigma-finite factors, the product measure exists, has the rectangle formula, is sigma-finite, and is unique

Statement

Let $(X, A, μ)$ and $(Y, B, ν)$ be sigma-finite measure spaces. Then:

the set function $E \mapsto (μ \times ν) (E)$ of The product measure of two sigma-finite measure spaces is a measure on $A \otimes B$ ;
for measurable rectangles, $(μ \times ν) (A \times B) = μ (A) ν (B),$ with the nonnegative extended-product convention $0 \cdot \infty = \infty \cdot 0 = 0$ ;
the measure $μ \times ν$ is sigma-finite; and
it is the unique measure on $A \otimes B$ with the rectangle formula.

Statement references

The product measure of two sigma-finite measure spaces

Facts & Assumptions

Given: Sigma-finite measure spaces $(X, A, μ)$ and $(Y, B, ν)$ .

[L1]

For every product-measurable set $E$ , $(μ \times ν) (E) = \int_{X} ν (E_{x}) d μ = \int_{Y} μ (E^{y}) d ν .$ (The product measure of two sigma-finite measure spaces)

[L2]

Monotone convergence passes increasing limits through nonnegative integrals. (Monotone convergence for the integral)

[L3]

A measure is determined by its values on a sigma-finite generating pi-system. (Measures agreeing on a generating pi-system are equal under an increasing finite-measure exhaustion from that pi-system)

[L4]

The nonnegative integral is additive for finite sums of nonnegative measurable functions. (Additivity of the nonnegative Lebesgue integral)

[L5]

The nonnegative integral agrees with the simple integral, whose zero coefficient on a null cell contributes zero even when another factor is infinite (The nonnegative integral agrees with the simple integral on simple functions, The integral of a nonnegative simple function).

[A1]

Measurable rectangles form a pi-system that generates $A \otimes B$ .

[A2]

Since $μ$ and $ν$ are sigma-finite, there are measurable exhaustions $X_{n} ↑ X$ and $Y_{n} ↑ Y$ with $μ (X_{n}), ν (Y_{n}) < \infty$ .

Proof

technique · direct

1.1L1L5

If $E = A \times B$ is a measurable rectangle, then $(A \times B)_{x} = {\begin{cases} B, & x \in A, \\ \emptyset, & x \notin A . \end{cases}$ Thus its section-measure function is $ν (B) 1_{A}$ , where the value outside $A$ is zero even when $ν (B) = \infty$ . The simple-function integral [L5] gives $(μ \times ν) (A \times B) = \int_{X} ν ((A \times B)_{x}) d μ = μ (A) ν (B)$ with $0 \cdot \infty = 0$ when $μ (A) = 0$ , and symmetrically when $ν (B) = 0$ . This proves the rectangle formula in every extended-value case.

1.2L1L2L4

Let $E_{1}, E_{2}, \dots$ be pairwise disjoint measurable subsets of $X \times Y$ , and put $F_{N} : = ⋃_{k \leq N} E_{k}$ . Then $(F_{N})_{x} = ⋃_{k \leq N} (E_{k})_{x}$ is a disjoint union, so $ν ((F_{N})_{x}) = \sum_{k \leq N} ν ((E_{k})_{x})$ for every $x$ . The section functions are nonnegative measurable, and finite additivity of their integrals gives $(μ \times ν) (F_{N}) = \int_{X} \sum_{k \leq N} ν ((E_{k})_{x}) d μ = \sum_{k \leq N} (μ \times ν) (E_{k}) .$ Since the section functions for $F_{N}$ increase pointwise to that of $⋃_{k \geq 1} E_{k}$ , [L2] gives $(μ \times ν) (⋃_{k \geq 1} E_{k}) = \sum_{k \geq 1} (μ \times ν) (E_{k}) .$ The empty set has product measure zero, so $μ \times ν$ is a measure.

2.1A2step 1.1

By [A2] and step 1.1, each rectangle $X_{n} \times Y_{n}$ has finite product measure $(μ \times ν) (X_{n} \times Y_{n}) = μ (X_{n}) ν (Y_{n}) < \infty,$ and $⋃_{n} (X_{n} \times Y_{n}) = X \times Y .$ Hence $μ \times ν$ is sigma-finite.

3.1A1L3step 1.1step 2.1∎

Let $ρ$ be another measure on $A \otimes B$ with the same rectangle formula. Step 1.1 shows that $ρ$ and $μ \times ν$ agree on the generating pi-system of measurable rectangles, and step 2.1 gives the required sigma-finite exhaustion. Therefore [L3] implies $ρ = μ \times ν$ on all of $A \otimes B$ . This proves existence, the rectangle formula, sigma-finiteness, and uniqueness.

For sigma-finite factors, the product measure exists, has the rectangle formula, is sigma-finite, and is unique

Statement

Statement references

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources