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 ${\displaystyle \mu }$ 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 ${\displaystyle \nu \ll \mu ,}$^[1] which is denoted as ${\displaystyle \mu \sim \nu .}$ That is, two measures are equivalent if they satisfy ${\displaystyle {\mathcal {N}}_{\mu }={\mathcal {N}}_{\nu }.}$

Examples

On the real line

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.

Abstract measure space

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.

Supporting measures

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 ${\displaystyle \mu .}$^[2]

References

1. Klenke, Achim (2008). Probability Theory. Berlin: Springer. p. 156. doi:10.1007/978-1-84800-048-3. ISBN 978-1-84800-047-6.
2. Kallenberg, Olav (2017). Random Measures, Theory and Applications. Switzerland: Springer. p. 21. doi:10.1007/978-3-319-41598-7. ISBN 978-3-319-41596-3.