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.

Definition. Let be a measurable space, and let be two measures. Then is said to be equivalent to if

for measurable sets in , 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 .

**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.

**Application**. Let be a finite measure on , and define

Show that is finite a.e. with respect to the Lebesgue measure on .

*Proof.* Let

then where

and

Clearly

and by Fubini’s Theorem

then

Since

Thus the following function

finite a.e. with respect to the measure . The conclusion follows from that the measure and the Lebesgue measure are equivalent.

Source: http://en.wikipedia.org/wiki/Equivalence_(measure_theory)

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