Equivalence (measure theory)

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

[edit] Definition

Let $(X, \mathcal{F})$ be a measure space, and let $\mu, \nu : \mathcal{F} \to [0, + \infty]$ be two measures. Then $μ$ is said to be equivalent to $ν$ if

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

for measurable sets $A \in \mathcal{F}$ . Equivalence is often denoted $\displaystyle{\mu \sim \nu}$ or $\mu \approx \nu$ .

In terms of absolute continuity of measures, two measures are equivalent if and only if each is absolutely continuous with respect to the other:

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

Equivalence of measures is an equivalence relation on the set of all measures $\mathcal{F} \to [0, + \infty].$

[edit] Examples

Gaussian measure and Lebesgue measure are equivalent to one another.
Lebesgue measure and Dirac measure are inequivalent.

[edit] References

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