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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

1.1L2L3given

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 $f$ .

2.1L1step 1.1

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.

3.1step 2.1algebra∎

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

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

Statement

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources