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 probability zero.

Definition[edit]

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:

\mu \sim \nu \iff \mu \ll \nu \ll \mu.

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: \mu(A) = 0 \iff \nu(A) = 0

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

Examples[edit]

References[edit]