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 $$\mu (A)=\underset{A}{\int }{\mathrm{𝟏}}_{[0,1]}(x)\mathrm{d}x$$ $$\nu (A)=\underset{A}{\int }{x}^2{\mathrm{𝟏}}_{[0,1]}(x)\mathrm{d}x$$ 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 Template:Visible anchor of a measure $\nu$ if $\mu$ is $\sigma$-finite and $\nu$ is equivalent to $\mu .$