setwise

Setwise is a term used in mathematics, primarily in measure theory and probability, to describe convergence or relationships defined with respect to sets. The prototypical usage is setwise convergence of a sequence of measures or probability distributions.

Definition: Let (Ω, F) be a measurable space and μ_n, μ be measures on F. We say μ_n converges setwise to μ if for every A ∈ F, μ_n(A) → μ(A). When μ_n are probability measures, this is equivalent to the distributions of corresponding random variables converging setwise.

Relationship to other convergences: Setwise convergence implies weak convergence (convergence of integrals against all bounded continuous functions) and hence convergence in distribution, but it is a stronger form of convergence than the usual weak convergence. It is strictly stronger than convergence in distribution in general, and does not imply almost sure convergence of the associated random variables. If μ_n converges in total variation to μ, then μ_n converges setwise to μ.

Non-measure contexts: The word setwise can also be used more broadly to describe operations or properties defined with regard to a set, but in mathematics it is most often encountered in the context described above.

Other uses: Setwise may appear as a brand or product name, but this article focuses on the mathematical sense of the term.