# Ngô Quốc Anh

## August 17, 2009

### The use of equivalence in measure

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.

1. Gaussian measure and Lebesgue measure on the real line are equivalent to one another.
2. 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.