Equivalence (measure theory)
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In mathematics, and specifically in measure theory, equivalence is a notion of two measures being "the same". Two measures are equivalent if they have the same null sets.
[edit] Definition
Let (X, Σ) be a measure space, and let μ, ν : Σ → [0, +∞] be two measures. Then μ is said to be equivalent to ν if
for measurable sets A in Σ, i.e. the two measures have precisely the same null sets. Equivalence is often denoted 
or 
.
In terms of absolute continuity of measures, two measures are equivalent if and only if each is absolutely continuous with respect to the other:
Equivalence of measures is an equivalence relation on the set of all measures Σ → [0, +∞].
[edit] 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.
