# Equivalence (measure theory)

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 $(X,{\mathcal {A}})$ , and let

${\mathcal {N}}_{\mu }:=\{A\in {\mathcal {A}}\mid \mu (A)=0\}$

and

${\mathcal {N}}_{\nu }:=\{A\in {\mathcal {A}}\mid \nu (A)=0\}$

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 $\mu$  iff ${\mathcal {N}}_{\nu }\supseteq {\mathcal {N}}_{\mu }$ . This is denoted as $\nu \ll \mu$ .

The two measures are called equivalent iff $\mu \ll \nu$  and $\nu \ll \mu$ , which is denoted as $\mu \sim \nu$ . That is, two measures are equivalent if they satisfy ${\mathcal {N}}_{\mu }={\mathcal {N}}_{\nu }$ .

## Examples

### On the real line

Define the two measures on the real line as

$\mu (A)=\int _{A}\mathbf {1} _{[0,1]}(x)\mathrm {d} x$
$\nu (A)=\int _{A}x^{2}\mathbf {1} _{[0,1]}(x)\mathrm {d} x$

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 $(X,{\mathcal {A}})$  and let $\mu$  be the counting measure, so

$\mu (A)=|A|$ ,

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, ${\mathcal {N}}_{\mu }=\{\emptyset \}$ . So by the second definition, any other measure $\nu$  is equivalent to the counting measure iff it also has just the empty set as the only $\nu$ -null set.

## Supporting measures

A measure $\mu$  is called a supporting measure of a measure $\nu$  if $\mu$  is $\sigma$ -finite and $\nu$  is equivalent to $\mu$ .