I was reading this question here
Fubini's Theorem double integral with sin and $e^{-x}$ but I do not know why are we allowed to replace the integral with respect to the product measure $\mu$ with iterated integrals? could anyone explain this for me please?
We have that $E=\{(x,y)\in \mathbb R^2:0\le y\le \sqrt x\}.\ $
If we can show that that $\int_E \left|\frac{y}{x} e^{-x}\sin x\right|\ d(m\times m)$ is finite, then the result follows by Fubini's theorem.
To do this, we split $E$ into a union of the two sets
$E_1=\{(x,y):0\le y\le \sqrt x;\ 0 \le x\le 1\}$ and $E_2=\{(x,y):0\le y\le \sqrt x;\ x\ge 1\}$.
Then, on $E_1,\ \left|\frac{y}{x} e^{-x}\sin x\right|\le 2ye^{-x}$, and the integral of this over $E_1$ is finite.
On $E_2,\ \left|\frac{y}{x} e^{-x}\sin x\right|\le \sqrt xe^{-x}$, and the integral of this function over $E_2$ is also finite.