Originally Posted by

**HappyJoe** Hello.

I haven't used Fubini's theorem for ages, and now that I came across a use of it, I am confused.

According to my notes, the following is justified by Fubini's theorem:

$\displaystyle \int_0^t\left(\int_s^t df(u)\right)dg(s) = \int_0^t\left(\int_{(0,u)}dg(s)\right)df(u).$

In my case, the lower limit of the integral is not included in the set over which we integrate, while the upper limit is, so for instance

$\displaystyle \int_0^t dg(s) = \int_{(0,t]}dg(s).$

Fubini's theorem states that under certain conditions, we have

$\displaystyle \int_A\left(\int_B f(x,y) dy\right) dx = \int_B\left(\int_A f(x,y) dx\right) dy,$

for some given pair of measures.

In my case, the set $\displaystyle B$ varies with $\displaystyle s$, which is a variable of integration in the outer integral. Upon the changing of the order of integration, why do the limits of integration appear as they do? It is not clear to me how to apply Fubini, when the sets $\displaystyle A$ and $\displaystyle B$ vary with the integration variables.