What is $dF(z)$ in the expected utility framework?

Question

Background: from a Microeconomics course,

$F$ is a cdf. In other words, if $F$ has a density function $f$, then $$F(z)={\int_{-\infty}^z f(x) dx} $$

Write the Bernoulli utility function $u: \mathbb R_+ \rightarrow \mathbb R$ such that the preference is represented by
$$U(F) = \int u(z) dF(z)$$

If $F$ has density $f$, then $U(F) = \int u(z) f(z)dz$

I am unfamiliar with the notation $dF(z)$, and don't quite understand what this means. Can someone help?

Answer 1

The last two lines of your quote tells you what $dF(z)$ is: $$ dF(z) = f(z)dz. $$

In your application, $f$ is the probability density on (point) events, $u$ is the utility, and the integral $U$ calculates the expected utility on subsets as a weighted average of point utilities with weights the probabilities.