---
tag: ABPK3RW5MW
kind: concept
name: Choquet theory
msc: [46A55, 52A07]
sources:
  - url: https://en.wikipedia.org/wiki/Choquet_theory
same_as:
  wikidata: Q1076274
  wikipedia: Choquet theory
---

## Summary

Choquet theory studies how a point of a compact [[convex set|02AS45T3H9]] is an average of its [[extreme points|3MSGFJM2NX]].
The average is taken with respect to a [[probability measure|059TJN0NZ1]] on the set, called a representing measure.
It refines the [[Krein–Milman theorem|RSRAY0Q68S]], which only says that the set is the closed convex hull of its extreme points.
The theory applies to sets of measures, of states and of positive functions, where extreme points are the simplest objects.

## Explanations

### L1 human

In a triangle, every point is a weighted average of the three corners, with unique weights.
Choquet theory extends this picture to infinite-dimensional convex sets, where a point may need infinitely many extreme points and a measure replaces the finite weights.
In a square, the centre is the average of two opposite corners in two different ways.
So the weights need not be unique, and the sets with unique weights, the simplices, play a special role.