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An integrable function whose sections vanish outside finite sets has multiple integral zero

Statement

Let $A \subseteq R^{p}$ and $B \subseteq R^{q}$ be nondegenerate closed rectangles, and let $f : A \times B \to R$ be Riemann integrable. If the set $S : = {x \in A : f_{x} is not identically 0}$ is finite, then $\int_{A \times B} f = 0$ . The analogous assertion holds with the coordinate blocks exchanged.

Facts & Assumptions

Given: An integrable $f : A \times B \to R$ whose nonzero $B$ -sections are indexed by a finite set $S$ .

[L1]

Riemann--Fubini permits a content-zero exceptional set of parameters and identifies the multiple integral with the resulting iterated integral (Riemann--Fubini on product rectangles, with lower and upper section integrals and content-zero exceptional sections).

[L2]

A set has content zero when it admits finite cube covers of arbitrarily small total volume (Measure zero and content zero in $R^{m}$ by countable and finite cube covers).

Proof

technique · direct

1.1L2given

A finite subset of $R^{p}$ has content zero by [L2]: for a given $ε > 0$ , cover its finitely many points by cubes whose total volume is below $ε$ .

2.1L1step 1.1

Outside $S$ every section is identically zero and has integral zero. Complete the section-integral function by the value $0$ on $S$ and apply [L1]; the resulting outer function is identically zero, so the multiple integral is zero.

3.1step 2.1algebra∎

If $S$ is empty then $f$ itself is identically zero, and step 2.1 still applies. Exchanging the coordinate blocks proves the symmetric assertion.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

13 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

J. Lebl, Basic Analysis II, Exercise 10.2.4 (standard reference, not scraped)