In mathematics, and specifically in measure theory, equivalence is a notion of two measures being qualitatively similar. Specifically, the two measures agree on which events have measure zero.

Definition

Let \mu and \nu be two measures on the measurable space and let and be the sets of \mu-null sets and \nu-null sets, respectively. Then the measure \nu is said to be absolutely continuous in reference to \mu if and only if This is denoted as The two measures are called equivalent if and only if \mu \ll \nu and which is denoted as That is, two measures are equivalent if they satisfy

Examples

On the real line

Define the two measures on the real line as for all Borel sets A. Then \mu and \nu are equivalent, since all sets outside of [0,1] have \mu and \nu measure zero, and a set inside [0,1] is a \mu-null set or a \nu-null set exactly when it is a null set with respect to Lebesgue measure.

Abstract measure space

Look at some measurable space and let \mu be the counting measure, so where |A| is the cardinality of the set a. So the counting measure has only one null set, which is the empty set. That is, So by the second definition, any other measure \nu is equivalent to the counting measure if and only if it also has just the empty set as the only \nu-null set.

Supporting measures

A measure *mu*** is**** called**** a **** of**** a measure *nu*** if**** *mu*** is**** *sig**ma**-fin**it**e*** and *nu*** is**** equivalent**** to**** *mu***.****