nLab σ-ideal

Definition

Suppose $X$ is a set and $M$ is a σ-algebra of subsets of $X$.

A σ-ideal of $M$ is a subset $N\subset M$ that is closed under countable unions and passage to subsets: if $a\in N$, $b\in M$, and $b\subset a$, then $b\in N$.

Example

If $\mu$ is a measure on a measurable space $(X,M)$, then

$$N_\mu = \{m\in M\mid \mu(m)=0\}$$

is a σ-ideal.

Applications

Sometimes we do not have a canonical measure $\mu$ at our disposal, but we do have a canonical σ-ideal of negligible sets. This is the case, for example, for smooth manifolds and locally compact groups.

Replacing the data of a measure $\mu$ with the data of a σ-ideal $N$ results in the concept of an enhanced measurable space $(X,M,N)$. See the article categories of measure theory for more details and motivation.

Related concepts

• σ-algebra

• enhanced measurable space

• categories of measure theory