Content

Builds on Semiring of sets Extended real number line

Definition

Let $X$ be a set and let $\mathcal{S}$ be a semiring of sets on $X$ . More generally, $\mathcal{S}$ may be any family of subsets of $X$ that contains $\varnothing$ . A content on $\mathcal{S}$ is a map $\mu \colon \mathcal{S} \to [0, \infty]$ , with values in the extended real line, such that:

$\mu(\varnothing) = 0$ .
If $A_1, \ldots, A_n \in \mathcal{S}$ are pairwise disjoint and $A_1 \cup \cdots \cup A_n \in \mathcal{S}$ , then $\mu(A_1 \cup \cdots \cup A_n) = \mu(A_1) + \cdots + \mu(A_n).$

Remarks

Example: on the semiring of bounded intervals $[a, b)$ in $\R$ , length $\mu([a, b)) = b - a$ is a content. Indeed, if $[a, b)$ is a disjoint union of finitely many such intervals, their lengths add up to $b - a$ . Non-example: on the same semiring, $\mu([a, b)) = (b - a)^2$ fails clause 2, since $[0, 2) = [0, 1) \cup [1, 2)$ but $4 \ne 1 + 1$ . On a ring of sets a content is monotone and finitely subadditive. A content on a ring is called a finitely additive measure by many authors. A content that is countably additive is a premeasure. The word content also names a set function used to build Haar measure. It is defined on the compact subsets of a space; it is monotone and finitely subadditive, and additive on disjoint compact sets.

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Tag	`JGF3WBT4GC`
URI	`m2://JGF3WBT4GC`
Kind	definition
Writing	W0 stub
MSC	28A12
Defines	Content
Uses	Semiring of sets, Extended real number line
Changed	2 November 2026

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Definition

Remarks

Reading path

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