Suppose $f(x,y)$ is a locally integrable function for $x\in\mathbb{R}^n$ and $y\in\mathbb{R}^m$ with $n,m\geq1$. Suppose $E$ is a measurable bounded subset of $\mathbb{R}^n\times\mathbb{R}^n$. For each $x\in\mathbb{R}^n$, we write $$E_x=\{y\in\mathbb{R}^m: (x,y)\in E\}. $$ Can we write the integral of $f$ over $E$ as $$\int_Ef(x,y)d(\mu\times\nu)(x,y)=\int_{\mathbb{R}^n}\int_{E_x}f(x,y)d\nu(y)d\mu(x)? $$ ($\mu$ and $\nu$ are the Lebesgue measures on $\mathbb{R}^n$ and $\mathbb{R}^n$ respectively)
Of course if $E$ was the product of two sets $F\times F'$ with $F\subset \mathbb{R}^n$ and $F'\subset\mathbb{R}^n$, the double integral would be over $F$ and $F'$ by Fubini's theorem. But the form of $E$ is not the product of two sets. There is something similar in multivariable calculus, but not as general as this.