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 $$ \int_\mathbb{R} \int_0^t f(g(x)) dx dg(x) = \int_0^t \int_\mathbb{R} f(g(x))dg(x)dx $$

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

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