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    • CommentRowNumber1.
    • CommentAuthorDmitri Pavlov
    • CommentTimeMay 3rd 2024

    Created:

    Definition

    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.

    Example

    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.

    Applications

    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.

    Related concepts

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