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

Statement

Let $E\subseteq\mathbb R^{p+q}$ be a bounded Jordan set and let $g:E\to\mathbb R$ be Riemann integrable. Suppose that for all $x$ outside a content-zero set $N\subseteq\mathbb R^p$ , 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 $\int_Eg=\int h(x)\,dx.$ The symmetric assertion holds for the other coordinate block.

Facts & Assumptions

Given: A bounded Jordan set $E$ , an integrable $g:E\to\mathbb R$ , 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 $A\subseteq\mathbb R^p$ and $B\subseteq\mathbb R^q$ 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 $\int_Eg=\int_Ah$ .

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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Direct dependencies and their dependencies through the next three levels: 86 results over 22 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.

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)