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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)

[L5]

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

Proof

technique · direct

1.1L2L5

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 by [L5] 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.

2.1L1L3L4step 1.1∎

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.

Depends on

Used by

Mass conservation and positivity of the heat flow Corollary
Polar integration may discard the cut locus Corollary
The Brownian square martingale Corollary
The graph of a measurable function Rⁿ to R is Lebesgue null Corollary
A gradient-only Poincare estimate needs normalisation Counterexample
Flat hyperplanes do not have spherical stationary-phase decay Counterexample
Iid strong law fails at infinite absolute mean Counterexample
Pointwise shock values do not affect the weak solution Counterexample
Product-measure equality is not pointwise equality Counterexample
The Carleson maximal operator is not strong type (1,1) Counterexample
The density ratio is undefined on zero marginal fibres Counterexample
The diagonal under Lebesgue times counting measure shows that Tonelli needs sigma-finiteness Counterexample
The local conservation law need not integrate to a finite conserved energy Counterexample
The ordinary chain rule fails for Brownian motion Counterexample
The unitary dual need not be Hausdorff Counterexample
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Locally square-integrable predictable Brownian integrands Definition
Logarithmic potential and energy of a positive compactly supported measure Definition
Newtonian potential of compactly supported data Definition
Period-one Fourier coefficients, partial sums, and convolution on the torus Definition
The Lusin area function for a fixed admissible kernel and aperture Definition
Adjoints of shifts, multiplication and integral operators Example
Brownian finite-dimensional density Example
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Hörmander estimate with a Gaussian weight Example
Integral of Brownian motion against itself Example
Ito formula for Brownian powers Example
Lowest K-types of the first holomorphic discrete series Example
Periodic derivative and its unitary translation group Example
Regular conditional law of one coordinate given another Example
The Hilbert transform of an H¹ atom is integrable Example
The square of the Volterra operator has zero trace Example
Tonelli and the geometric series compute int₀¹ int₀¹ 1/(1-xy) dx dy = pi²/6 Example
Tonelli and the plane polar formula give int_R e^-x² dx = sqrt(pi) Example
Two Cantor sets with different logarithmic capacities Example
Volterra operator is Hilbert Schmidt and quasinilpotent Example

…and 115 more results.

Dependency tree · two levels

31 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

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)