## Double and Iterated Integrals

Let and be two -finite measure spaces, and let be the product measure on .

If is a function on so that its integral is defined — either is integrable, or its integral diverges definitely — then we write it as any of

and call it the “double integral” of over . We can also consider the sections and . For any given , we set

if the integral exists. If the resulting function is integrable, then we write

The latter notation, with the measure before the integrand is less common, but it can be seen in older texts. I’ll usually stick to the other order.

Similarly, we can define the function as the integral of the -section if it exists. If is integrable, we write

where, again, the latter notation is deprecated. These integrals are called the “iterated integrals” of . We can also define double and iterated integrals over a measurable subset , as usual, and write

[…] if is a non-negative measurable function on , then we have the following equalities between the double integral and the two iterated […]

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