set function

Definition

A set function on $\Omega$ is a map $\mu :A\to Y$ where whose domain is of sets:

$$A\subseteq \mathcal{P}(\Omega )=\{S|S\subseteq \Omega \}$$

Set function

A set function is a function whose inputs are sets from some fixed collection (subsets of $\Omega$ denoted by the power set):

$$\varphi :\mathcal{F}\to Y,\mathcal{F}\subseteq 2^{\Omega }$$

for some codomain $Y$ which can be any set.

In contrast, a set-valued function takes points as input and returns sets.