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 $(X,{𝒜}),$ and let $${{𝒩}}_{\mu }:=\{A\in {𝒜}\mid \mu (A)=0\}$$ and $${{𝒩}}_{\nu }:=\{A\in {𝒜}\mid \nu (A)=0\}$$ 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 ${{𝒩}}_{\nu }\supseteq {{𝒩}}_{\mu }.$ 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 ${{𝒩}}_{\mu }={{𝒩}}_{\nu }.$

Examples

On the real line

Define the two measures on the real line as $$\mu (A)=\underset{A}{\int }{\mathbf{𝟏}}_{[0,1]}(x)\mathrm{d}x$$ $$\nu (A)=\underset{A}{\int }{x}^2{\mathbf{𝟏}}_{[0,1]}(x)\mathrm{d}x$$ 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 $(X,{𝒜})$ 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, ${{𝒩}}_{\mu }=\{\varnothing \}.$ 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 $\mu .$