<p>Not all cdf’s have a density function, (for example if <span class="math-container">$F$</span> is not differentiable). However, when they do have a density, the notation <span class="math-container">$dF(z)$</span> is equivalent to <span class="math-container">$f(z)dz$</span>. When performing integrals. However, even if the density does not exists, you can still write the expectation using the notation <span class="math-container">$dF(z)$</span>. 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.</p>

CC BY-SA 4.0
2019-05-21T06:12:00.443Z
This notation also allows to encompass the case of discrete random variables, in which case $$U(F) = \int u(z) dF(z)=\sum_k u(z_k) p(z_k) $$
Bertrand
CC BY-SA 4.0
2019-05-21T07:05:23.177Z
And the method is called [Riemann-Stieltjes integral](https://en.wikipedia.org/wiki/Riemann%E2%80%93Stieltjes_integral).
Giskard
CC BY-SA 4.0
2019-05-21T05:01:20.753Z
2019-05-21T07:03:46.317Z
2019-05-21T07:03:46.317Z
Regio
<p>Background: from a Microeconomics course,</p>

<blockquote>
  <p><span class="math-container">$F$</span> is a cdf. In other words, if <span class="math-container">$F$</span> has a density function <span class="math-container">$f$</span>, then 
  <span class="math-container">$$F(z)={\int_{-\infty}^z f(x) dx}  $$</span></p>
  
  <p>Write the Bernoulli utility function <span class="math-container">$u: \mathbb R_+ \rightarrow \mathbb R$</span> such that the preference is represented by<br>
  <span class="math-container">$$U(F) = \int u(z) dF(z)$$</span></p>
  
  <p>If <span class="math-container">$F$</span> has density <span class="math-container">$f$</span>, then <span class="math-container">$U(F) = \int u(z) f(z)dz$</span></p>
</blockquote>

<p>I am unfamiliar with the notation <span class="math-container">$dF(z)$</span>, and don't quite understand what this means. Can someone help?</p>

<microeconomics><utility><expected-utility><definition>
Meaning of $dF(z)$ in expected utility framework
Yejin
# Meaning of $dF(z)$ in expected utility framework

(Asked by: username_0 on 2019-05-21)

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?

**Tags:** microeconomics, utility, expected-utility, definition


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## 1 answers

### Answer by username_1 on 2019-05-21. Score: 7 | Accepted answer

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.

> **Comments:**
> **username_2**: This notation also allows to encompass the case of discrete random variables, in which case $$U(F) = \int u(z) dF(z)=\sum_k u(z_k) p(z_k) $$  
> **username_3**: And the method is called [Riemann-Stieltjes integral](https://en.wikipedia.org/wiki/Riemann%E2%80%93Stieltjes_integral).  

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