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Let $U\subset\mathbb R^m, V\subset\mathbb R^n$ be open sets and $f:U\times V\to \mathbb R$ a continuous function. I want to prove

$$\int_Udx\int_Vf(x,y)dy=\int_Vdy\int_Uf(x,y)dx=\int_{U\times V}f$$

If the integral $\int_{U\times V}f$ is convergent.

This is a kind of a generalization of the repeated integral theorem which uses blocks instead of open sets (these open sets can be bounded or unbounded).

If these open sets are bounded we cover these ones with blocks and apply the repeated integral theorem. I don't know how to proceed if these open sets are unbounded.

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    Are your open sets also dense? – 2017-01-28
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    @N74 no necessarily – 2017-01-28
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    @N74 why? do you have some ideas how to solve this question? – 2017-01-29
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    Unfortunately I don't know how to handle integrals on non-dense sets. – 2017-01-29

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