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

The two measures are called equivalent if and only if $\mu \ll \nu$ 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, ${\mathcal {N}}_{\mu }=\{\varnothing \}.$ 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 supporting measure of a measure $\nu$ if $\mu$ is $\sigma$-finite and $\nu$ is equivalent to $\mu .$