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1. • CommentRowNumber1.
   • CommentAuthorDmitri Pavlov
   • CommentTimeMay 3rd 2024
   • PermaLink
   Author: Dmitri Pavlov
   Format: MarkdownItexCreated: ## 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]] <a href="https://ncatlab.org/nlab/revision/%CF%83-ideal/1">v1</a>, <a href="https://ncatlab.org/nlab/show/%CF%83-ideal">current</a>

   Created:

   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

   v1, current