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

The two measures are called equivalent if and only if ${\displaystyle \mu \ll \nu }${\displaystyle \mu \ll \nu } and ${\displaystyle \nu \ll \mu ,}${\displaystyle \nu \ll \mu ,}^[1] which is denoted as ${\displaystyle \mu \sim \nu .}${\displaystyle \mu \sim \nu .} That is, two measures are equivalent if they satisfy ${\displaystyle {\mathcal {N}}_{\mu }={\mathcal {N}}_{\nu }.}${\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 \mu (A)=\int _{A}\mathbf {1} _{[0,1]}(x)\mathrm {d} x} $${\displaystyle \nu (A)=\int _{A}x^{2}\mathbf {1} _{[0,1]}(x)\mathrm {d} x}$${\displaystyle \nu (A)=\int _{A}x^{2}\mathbf {1} _{[0,1]}(x)\mathrm {d} x} for all Borel sets ${\displaystyle A.}${\displaystyle A.} Then ${\displaystyle \mu }${\displaystyle \mu } and ${\displaystyle \nu }${\displaystyle \nu } are equivalent, since all sets outside of ${\displaystyle [0,1]}${\displaystyle [0,1]} have ${\displaystyle \mu }${\displaystyle \mu } and ${\displaystyle \nu }${\displaystyle \nu } measure zero, and a set inside ${\displaystyle [0,1]}${\displaystyle [0,1]} is a ${\displaystyle \mu }${\displaystyle \mu }-null set or a ${\displaystyle \nu }${\displaystyle \nu }-null set exactly when it is a null set with respect to Lebesgue measure.

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

A measure ${\displaystyle \mu }${\displaystyle \mu } is called a supporting measure of a measure ${\displaystyle \nu }${\displaystyle \nu } if ${\displaystyle \mu }${\displaystyle \mu } is ${\displaystyle \sigma }${\displaystyle \sigma }-finite and ${\displaystyle \nu }${\displaystyle \nu } is equivalent to ${\displaystyle \mu .}${\displaystyle \mu .}^[2]

• Equivalence (mathematics)
• Measure theory