Equivalence (measure theory)

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

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

• Gaussian measure and Lebesgue measure on the real line are equivalent to one another.
• Lebesgue measure and Dirac measure on the real line are inequivalent.

References

• Halmos, Paul R. (1974). Measure Theory. Springer. p. 126. ISBN 0-387-90088-8.