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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

1.1

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 $ε$ .

L2given

2.1

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.

L1step 1.1

3.1

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

step 2.1algebra∎

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)