nLab σ-ideal


Suppose XX is a set and MM is a σ-algebra of subsets of XX.

A σ-ideal of MM is a subset NMN\subset M that is closed under countable unions and passage to subsets: if aNa\in N, bMb\in M, and bab\subset a, then bNb\in N.


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

N μ={mMμ(m)=0}N_\mu = \{m\in M\mid \mu(m)=0\}

is a σ-ideal.


Sometimes we do not have a canonical measure μ\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 μ\mu with the data of a σ-ideal NN results in the concept of an enhanced measurable space (X,M,N)(X,M,N). See the article categories of measure theory for more details and motivation.

Created on May 3, 2024 at 03:30:46. See the history of this page for a list of all contributions to it.