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?


1 Answer 1


Not all cdf’s have a density function, (for example if $F$ is not differentiable). However, when they do have a density, the notation $dF(z)$ is equivalent to $f(z)dz$. When performing integrals. However, even if the density does not exists, you can still write the expectation using the notation $dF(z)$. The details of what it actually means and the subtle differences can be learned in any measure theory course. But hopefully this short explanation helps you not freak out every time you see that notation.

  • 4
    $\begingroup$ This notation also allows to encompass the case of discrete random variables, in which case $$U(F) = \int u(z) dF(z)=\sum_k u(z_k) p(z_k) $$ $\endgroup$
    – Bertrand
    Commented May 21, 2019 at 6:12
  • 3
    $\begingroup$ And the method is called Riemann-Stieltjes integral. $\endgroup$
    – Giskard
    Commented May 21, 2019 at 7:05

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