Definition of mean as an integral over the CDF I'm reading a statistics textbook which defines the mean of a random variable $X$ with CDF $F$ as a statistical function $t(\centerdot)$, where
$$ t(F) = \int x \, dF(x).$$
Can someone explain this definition? I'm familiar with the definition of the mean as an integral over the PDF $f$:
$$ \int x \, f(x) \, dx.$$
But what does it mean to have the function $F(x)$ in the variable of integration?

The cumulative density function $F$ is related to the probability density function $f$ by $dF(x)/dx=f(x)$. The equation you have in terms of $F$ can be re-expressed in terms of $f$ by substituting in $dF(x) = f(x)\,dx$. In fact, for many purposes, you can take this as the definition of the differential term $dF$. However, in more general circumstances where $F$ is not differentiable and the PDF $f$ is not well-defined, the form involving $dF$ still holds (interpreting it as a Riemann-Stieltjes integral). For example, if the distribution is discrete, so that $F$ is piecewise constant, then $dF$ becomes a sum over Dirac distributions.
