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Fubini over a bounded Jordan set when all but a content-zero family of sections are integrable

Statement

Let $E \subseteq R^{p + q}$ be a bounded Jordan set and let $g : E \to R$ be Riemann integrable. Suppose that for all $x$ outside a content-zero set $N \subseteq R^{p}$ , the section $E_{x}$ is Jordan measurable and $g_{x}$ is integrable over it. Put $h (x) = \int_{E_{x}} g_{x}$ there and assign arbitrary bounded values to $h$ on $N$ , with empty-section integral equal to $0$ . Then $h$ is integrable on any rectangle containing the projection of $E$ , its integral is independent of that rectangle and of the values on $N$ , and $\int_{E} g = \int h (x) d x .$ The symmetric assertion holds for the other coordinate block.

Facts & Assumptions

Given: A bounded Jordan set $E$ , an integrable $g : E \to R$ , and the stated content-zero exceptional family of sections.

[L1]

Riemann--Fubini applies to bounded functions on a product rectangle using lower and upper section integrals and a content-zero exceptional set (Riemann--Fubini on product rectangles, with lower and upper section integrals and content-zero exceptional sections).

[L2]

An empty Jordan section has integral zero, and section integrals are taken after zero extension to a bounding rectangle (Sections, lower and upper section integrals, and iterated Riemann integrals on product rectangles and Jordan sets).

[L3]

The Jordan-set integral is independent of the chosen bounding rectangle (The Riemann integral over a Jordan set is independent of the bounding rectangle).

Proof

technique · direct

1.1

Choose nondegenerate rectangles $A \subseteq R^{p}$ and $B \subseteq R^{q}$ with $E \subseteq A \times B$ , and extend $g$ by zero to $A \times B$ . For every $x$ , the resulting section is the zero extension to $B$ of $g_{x}$ , and it is identically zero when $E_{x}$ is empty.

L2given

2.1

The zero extension is integrable by the definition of the Jordan-set integral. Apply [L1]; outside $N$ its ordinary section integral is exactly $h (x)$ , so the exceptional-section clause gives $\int_{E} g = \int_{A} h$ .

L1step 1.1

3.1

Enlarging $A$ or $B$ only adds zero to the zero extension. Independence of the Jordan integral from a bounding rectangle [L3] and the content-zero invariance in [L1] therefore prove independence of both factor rectangles and of the assigned values on $N$ .

L1L2L3step 2.1algebra∎

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Sources

A. Leibman, Multidimensional Real Analysis, Theorem 5.4.2 (standard reference, not scraped)
J. Lebl, Basic Analysis II, §10.5 (standard reference, not scraped)