---
tag: 13F6B3BGA4
kind: concept
name: Set-valued measure
msc: [28B20]
sources:
  - url: https://en.wikipedia.org/wiki/Set-valued_function
---

## Summary

A set-valued measure assigns to each measurable set a nonempty subset of $\R^n$, and it is countably additive for the Minkowski sum of sets.
It generalises a [[vector measure|PS28BSAJKJ]], which assigns a single vector to each set.
Set-valued measures arise, for example, as integrals of set-valued functions over measurable sets.
They are studied in mathematical economics and in control theory.

## Explanations

### L1 human

An ordinary measure gives each set one number, its size.
A set-valued measure gives each set a whole region of possible values.
For two disjoint sets, the region of the union consists of all sums of a value from the first region and a value from the second.
For example, giving each set $A$ the interval $[0, \mu(A)]$ for a finite measure $\mu$ is a set-valued measure on the line.