Not signed in

Want to take part in these discussions? Sign in if you have an account, or apply for one below

• Username
• Password
• Remember me

• Sign in using OpenID
• Remember me

• Forgot your password?
• Apply for membership

2-category 2-category-theory abelian-categories adjoint algebra algebraic algebraic-geometry algebraic-topology analysis analytic-geometry arithmetic arithmetic-geometry book bundles calculus categorical categories category category-theory chern-weil-theory cohesion cohesive-homotopy-type-theory cohomology colimits combinatorics complex complex-geometry computable-mathematics computer-science constructive cosmology deformation-theory descent diagrams differential differential-cohomology differential-equations differential-geometry digraphs duality elliptic-cohomology enriched fibration foundation foundations functional-analysis functor gauge-theory gebra geometric-quantization geometry graph graphs gravity grothendieck group group-theory harmonic-analysis higher higher-algebra higher-category-theory higher-differential-geometry higher-geometry higher-lie-theory higher-topos-theory homological homological-algebra homotopy homotopy-theory homotopy-type-theory index-theory integration integration-theory internal-categories k-theory lie-theory limits linear linear-algebra locale localization logic mathematics measure measure-theory modal modal-logic model model-category-theory monad monads monoidal monoidal-category-theory morphism motives motivic-cohomology nlab noncommutative noncommutative-geometry number-theory of operads operator operator-algebra order-theory pages pasting philosophy physics pro-object probability probability-theory quantization quantum quantum-field quantum-field-theory quantum-mechanics quantum-physics quantum-theory question representation representation-theory riemannian-geometry scheme schemes set set-theory sheaf simplicial space spin-geometry stable-homotopy-theory stack string string-theory superalgebra supergeometry svg symplectic-geometry synthetic-differential-geometry terminology theory topology topos topos-theory tqft type type-theory universal variational-calculus

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