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A continuous function on a closed rectangle has repeated Riemann integrals in every coordinate order, all equal to its multiple integral

Statement

Let $Q = \prod_{j < n} [a_{j}, b_{j}] \subseteq R^{n}$ , where $n \geq 1$ and every $a_{j} < b_{j}$ . If $f : Q \to R$ is continuous, then for every permutation of the coordinates the corresponding repeated Riemann integral exists and equals $\int_{Q} f$ .

Facts & Assumptions

Given: A continuous real function $f$ on a nondegenerate closed rectangle $Q \subseteq R^{n}$ .

[L1]

Riemann--Fubini identifies the multiple integral with either iterated integral whenever the ordinary sections exist away from a content-zero exceptional set (Riemann--Fubini on product rectangles, with lower and upper section integrals and content-zero exceptional sections).

[L2]

Every continuous real function on a closed nondegenerate rectangle in positive dimension is Riemann integrable (Every continuous function on a closed nondegenerate rectangle in $R^{m}$ is Riemann integrable).

[L3]

A continuous map from a compact metric space to a metric space is uniformly continuous (Heine-Cantor: a continuous map from a compact metric space to any metric space is uniformly continuous).

Proof

technique · direct

1.1

Every coordinate section is continuous, hence integrable by [L2]. Moreover $f$ is uniformly continuous on the compact rectangle by [L3]; therefore integrating in one coordinate produces a continuous function of the remaining coordinates, since the difference of two section integrals is bounded by the interval length times the uniform oscillation of $f$ .

L2L3given

2.1

Apply [L1] to the first coordinate in a prescribed order. Step 1.1 makes the resulting function continuous, so the same argument applies to the next coordinate. Induction through the finite coordinate list gives the repeated integral.

L1step 1.1

3.1

For $n = 1$ the repeated integral is the original integral. At every later stage [L1] preserves its value, so every coordinate order gives $\int_{Q} f$ .

step 2.1algebra∎

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Sources

J. Lebl, Basic Analysis II, corollary after Theorem 10.2.3 (standard reference, not scraped)
A. Leibman, Multidimensional Real Analysis, §5.4 (standard reference, not scraped)