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Tonelli's theorem for nonnegative measurable functions on a sigma-finite product

Statement

Let $(X, A, μ)$ and $(Y, B, ν)$ be sigma-finite measure spaces, and let $f : X \times Y \to [0, \infty]$ be product-measurable. Then $x \mapsto \int_{Y} f_{x} d ν$ and $y \mapsto \int_{X} f^{y} d μ$ are measurable, and

$\int_{X \times Y} f d (μ \times ν) = \int_{X} (\int_{Y} f_{x} d ν) d μ = \int_{Y} (\int_{X} f^{y} d μ) d ν .$

Facts & Assumptions

Given: Sigma-finite measure spaces $(X, A, μ)$ and $(Y, B, ν)$ , and a product-measurable function $f : X \times Y \to [0, \infty]$ .

[L1]

Sections of a product-measurable function are measurable. (Every section of a product-measurable function is measurable)

[L2]

For a measurable set $E$ , the indicator function satisfies $\int_{X \times Y} 1_{E} d (μ \times ν) = \int_{X} ν (E_{x}) d μ = \int_{Y} μ (E^{y}) d ν .$ (For sigma-finite measures, the two section-measure integrals of a measurable set agree)

[L3]

Every nonnegative measurable function admits an increasing sequence of nonnegative simple functions converging pointwise to it. (Every nonnegative measurable function admits an explicit increasing sequence of simple approximations)

[L4]

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

Proof

technique · direct

1.1

If $s$ is a nonnegative simple function, write $s = \sum_{j = 1}^{m} c_{j} 1_{E_{j}}$ with $c_{j} \geq 0$ and measurable sets $E_{j}$ . Applying [L2] to each indicator $1_{E_{j}}$ and summing yields $\int s d (μ \times ν) = \int_{X} (\int_{Y} s_{x} d ν) d μ = \int_{Y} (\int_{X} s^{y} d μ) d ν .$ The inner integral functions are measurable because the same is true for each $1_{E_{j}}$ and simple combinations preserve measurability.

2.1

Choose simple functions $s_{n} ↑ f$ by [L3]. Then for each $x$ and $y$ one has $(s_{n})_{x} ↑ f_{x}$ and $(s_{n})^{y} ↑ f^{y}$ , so [L4] gives $\int_{Y} (s_{n})_{x} d ν ↑ \int_{Y} f_{x} d ν, \int_{X} (s_{n})^{y} d μ ↑ \int_{X} f^{y} d μ .$ Applying [L4] once more to the equalities of step 1.1 yields the stated measurability and the equality of all three integrals.

L1L3L4step 1.1∎

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Sources

Terence Tao, An Introduction to Measure Theory, Theorem 1.7.18 (standard reference, not scraped)
Gerald B. Folland, Real Analysis, 2nd ed., Theorem 2.37 (standard reference, not scraped)