Want to take part in these discussions? Sign in if you have an account, or apply for one below
Vanilla 1.1.10 is a product of Lussumo. More Information: Documentation, Community Support.
Created:
Suppose X is a set and M is a σ-algebra of subsets of X.
A σ-ideal of M is a subset N⊂M that is closed under countable unions and passage to subsets: if a∈N, b∈M, and b⊂a, then b∈N.
If μ is a measure on a measurable space (X,M), then
Nμ={m∈M∣μ(m)=0}is a σ-ideal.
Sometimes we do not have a canonical measure μ 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 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.
1 to 1 of 1