Changing the order of integration (for Lebesgue-Stieltjes integral and Riemann integral)

Question

Do the Lebesgue-Stieltjes integral and the Riemann integral have the same rules about the change of order of integration? I mean I know how to deal with Riemann integral, but I'm not sure if I can simply apply the same rules to the Lebesgue-Stieltjes.

Thanks.

It's double integration in $\mathbb{R}^2$, by the way.

Answer 1

The Fubini–Tonelli theorems apply to arbitrary product measure spaces (not just $\mathbb R^2$) as long as the factor spaces are both $\sigma$-finite.

Basically, if $(X,\mathscr M,\mu)$ and $(Y,\mathscr N,\nu)$ are $\sigma$-finite, then we can consider the product measure space $(X\times Y,\mathscr M\otimes\mathscr N,\mu\times\nu)$. We have the following:

• (Tonelli) If $f:X\times Y\to\mathbb [0,\infty]$ is $\mathscr M\otimes\mathscr N$-measurable, then the order of integrals can be interchanged: \begin{align*}&\int_{x\in X}\int_{y\in Y}f(x,y)\,\mathrm d\nu(y)\,\mathrm d\mu(x)=\int_{y\in Y}\int_{x\in X}f(x,y)\,\mathrm d\mu(x)\,\mathrm d\nu(y)\\=&\int_{(x,y)\in X\times Y}f(x,y)\,\mathrm d(\mu\times\nu)(x,y).\tag{*}\end{align*}
• (Fubini) If $f:X\times Y\to\mathbb C$ is $\mathscr M\otimes\mathscr N$-measurable and $$\int_{(x,y)\in X\times Y}|f(x,y)|\,\mathrm d(\mu\times\nu)(x,y)<\infty,$$ then every term appearing in $(*)$ above is well-defined, and the equalities there all hold.