Equivalence (measure theory)

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In mathematics, and specifically in measure theory, equivalence is a notion of two measures being qualitatively similar. Specifically, the two measures agree on which events have measure zero.


Let (X, Σ) be a measurable space, and let μ, ν : Σ → R be two signed measures. Then μ is said to be equivalent to ν if and only if each is absolutely continuous with respect to the other. In symbols:

Thus, any event A is a null event with respect to μ, if and only if it is a null event with respect to ν:

Equivalence of measures is an equivalence relation on the set of all measures Σ → R.