= Solution
Consider the <Borel set>
$$
D=\{(x,y):a<x\leq y\leq b\}.
$$
Its <indicator function> is nonnegative and <measurable>, so the <Tonelli theorem> applies to the <product measure>, without needing any separate absolute-<integrability> argument. At a fixed $y\in(a,b]$, its section in the first coordinate is $(a,y]$; outside $(a,b]$ that section is empty. Therefore
$$
\begin{aligned}
(\mu\otimes\nu)(D)
&=\int_{\mathbb R}\left(\int_{\mathbb R}\mathbf1_D(x,y)\,\mu(dx)\right)\nu(dy)\\
&=\int_{(a,b]}\mu((a,y])\,\nu(dy)\\
&=\int_{(a,b]}\bigl(F_\mu(y)-F_\mu(a)\bigr)\,\nu(dy).
\end{aligned}
$$
The last equality uses the definition of a <cumulative distribution function> on the half-closed <interval>. It retains all <point atoms of a measure> on the diagonal $x=y$, excludes a possible <point atom of a measure> at $x=a$, and includes $y=b$, exactly as required by the <set> $D$. If $a=b$, both sides vanish. This verifies the identity even when the <probability measures> are not <non-atomic measures>.<Probability: Theory and Examples (fifth edition) by Rick Durrett>[§1.7, Theorem 1.7.2, printed pp. 38–40: integration of a nonnegative measurable function on a product space]{ref}
Back to article page
