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 .
- 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 .
and by Fubini’s Theorem
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.