# Riemann-Stieltjes Integral - Changing Order of Integration

I am new to Riemann-Stieltjes integral. I want to ask a very basic question regarding changing the order of integration.

Let $$t > 0$$ and I have an integral that looks like this $$\int_\mathbb{R} \int_0^t f(g(x)) dx dg(x).$$

What is the condition so that I change the order of integration? Or mathematically we could write the integral like this $$\int_\mathbb{R} \int_0^t f(g(x)) dx dg(x) = \int_0^t \int_\mathbb{R} f(g(x))dg(x)dx$$

• I believe we still need the same assumptions Fubini requires. Is $f$ a functoom of a single variable? – Sean Roberson Dec 7 '18 at 4:44
• Function $f$ is a function of $g(x)$ and $x$. For example, the simplest one is $f(x) = g(x) + x$. – Ben Dec 7 '18 at 16:23