What is the $dF(X)$ in some integrals concerning probability densities? On multiple occasions I've come across statements about integrals of 
Considers the problem of minimizing the risk functional (Vapnik's Statistical Learning Theory) 
$$ R(\alpha ) = \int Q(z, \alpha ) dF(z) $$
Is this the same as having the probability density $F_Z(z)$ multiplied by a differential of the random variable $dz$ like this?
$$ R(\alpha ) = \int Q(z, \alpha ) f_Z(z) dz $$
I've read somewhere that $dF(z)$ is the probability measure so why is this different from the second formulation. How should I think about $dF(z)$?
I've seen works 
 A: This notation refers to the Lebesgue-Stieltjes integral and $F$ is the cumulative distribution function of the random variable under consideration.  This integral form is a useful way to write expectations of functions of random variables in cases where you want to include both continuous and discrete cases (and mixed cases).  In the event that $Z$ is a continuous random variable with density function $f_Z$ we obtain:
$$R(\alpha) = \int Q(z, \alpha) dF(z) = \int Q(z, \alpha) f_Z(z) dz.$$
In the event that $Z$ is a discrete random variable with mass function $f_Z$ we obtain:
$$R(\alpha) = \int Q(z, \alpha) dF(z) = \sum_z Q(z, \alpha) f_z(z).$$
If $dF$ is to be considered an object of its own (as opposed to just being a component of the notation of the Lebesgue-Stieltjes integral) then its meaning must be established by context and usage.  It usually refers to an infinitesimal induced by the cumulative distribution function $F$, which is technically a linear map.  With a slight abuse of notation it might refer to a measure, depending on context and usage.  What is certainly true is that the cumulative distribution function $F$ induces a unique probability measure for $X$ (this follows from the famous Carathéodory extension theorem) and we can write the Lebesgue-Stieltjes integral as a Lebesgue integral with respect to that measure.  It is usual to denote this measure with some standard measure notation such as $\mu_Z$, $\mathbb{P}_Z$, etc.  I suppose it is possible that some texts might use the notation $dF$ for this measure, but that would not be standard notation, and it would be confusing.
