Equivalence (measure theory)

In mathematics, and specifically in measure theory, equivalence is a notion of two measures being "the same".

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:

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

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 0387900888.