---
tag: PFF8SV6ENB
kind: concept
name: Content
msc: [28A12]
sources:
  - url: https://en.wikipedia.org/wiki/Content_(measure_theory)
same_as:
  wikidata: Q5165061
  wikipedia: Content (measure theory)
---

## Summary

A content is a size function on a family of sets that is only finitely additive, not countably additive.
It is the same idea as a [[finitely additive measure|KFWQ24D9M6]], usually on a [[ring of sets|KWE1DBMN9B]] or a [[semiring of sets|PYH82FMXQ2]].
Contents are the first step toward measures: a content that is also countably additive is a [[premeasure|PV0XXSWFC7]].
Jordan measure is a classical content that is not a measure.

## Explanations

### L1 human

A content gives a size to some sets, so that the size of two disjoint pieces together is the sum of their sizes.
It says nothing about cutting a set into infinitely many pieces.
For example, the area of finite unions of rectangles in the plane is a content.
The [[Jordan measure|0KTGJW1ZTH]] of a bounded region, defined by inner and outer approximation by rectangles, is another one.