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.

Let ${\displaystyle \mu }$and ${\displaystyle \nu }$be two measures on the measurable space ${\displaystyle (X,{\mathcal {A}}),}$and let $${\displaystyle {\mathcal {N}}_{\mu }:=\{A\in {\mathcal {A}}\mid \mu (A)=0\}}$$and $${\displaystyle {\mathcal {N}}_{\nu }:=\{A\in {\mathcal {A}}\mid \nu (A)=0\}}$$be the sets of ${\displaystyle \mu }$-null sets and ${\displaystyle \nu }$-null sets, respectively. Then the measure ${\displaystyle \nu }$is said to be absolutely continuous in reference to ${\displaystyle \mu }$if and only if ${\displaystyle {\mathcal {N}}_{\nu }\supseteq {\mathcal {N}}_{\mu }.}$This is denoted as ${\displaystyle \nu \ll \mu .}$

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

Define the two measures on the real line as $${\displaystyle \mu (A)=\int _{A}\mathbf {1} _{[0,1]}(x)\mathrm {d} x}$$$${\displaystyle \nu (A)=\int _{A}x^{2}\mathbf {1} _{[0,1]}(x)\mathrm {d} x}$$for all Borel sets ${\displaystyle A.}$Then ${\displaystyle \mu }$and ${\displaystyle \nu }$are equivalent, since all sets outside of ${\displaystyle [0,1]}$have ${\displaystyle \mu }$and ${\displaystyle \nu }$measure zero, and a set inside ${\displaystyle [0,1]}$is a ${\displaystyle \mu }$-null set or a ${\displaystyle \nu }$-null set exactly when it is a null set with respect to Lebesgue measure.

Look at some measurable space ${\displaystyle (X,{\mathcal {A}})}$and let ${\displaystyle \mu }$be the counting measure, so $${\displaystyle \mu (A)=|A|,}$$where ${\displaystyle |A|}$is the cardinality of the set a. So the counting measure has only one null set, which is the empty set. That is, ${\displaystyle {\mathcal {N}}_{\mu }=\{\varnothing \}.}$So by the second definition, any other measure ${\displaystyle \nu }$is equivalent to the counting measure if and only if it also has just the empty set as the only ${\displaystyle \nu }$-null set.

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