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