Sigma-additive set function

A σ-additive set function maintains additivity for countably infinite sets

A σ-additive set function is defined as a function that retains the additivity property even when dealing with an infinite collection of sets. This characteristic distinguishes σ-additive set functions from finitely additive set functions, which only guarantee additivity for a finite number of sets. The concept of σ-additivity is crucial in measure theory as it allows for the consistent definition of measures over infinite collections of sets.

Example

Consider the set of all natural numbers N. A σ-additive set function μ assigns a measure to each subset of N. If A and B are disjoint subsets of N, then μ(A ∪ B) = μ(A) + μ(B), regardless of whether A and B are finite or infinite in size.

Understanding σ-additivity is essential for working with measures in infinite contexts, such as probability theory and real analysis.

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