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Propositions from Fubini and iterated integrals

All propositions from Fubini and iterated integrals.

[PROPOSITION]

If $g$ is integrable on $[a,b]$ and $h$ is integrable on $[c,d]$ , then $f(x,y)=g(x)h(y)$ is integrable on $R=[a,b]\times[c,d]$ and

\iint_R g(x)h(y)\,dA = \left(\int_a^b g(x)\,dx\right) \left(\int_c^d h(y)\,dy\right).

The factor that does not involve the inner variable pulls out of the inner integral.