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 ${\displaystyle \mu }$  iff ${\displaystyle {\mathcal {N}}_{\nu }\supseteq {\mathcal {N}}_{\mu }}$ . This is denoted as ${\displaystyle \nu \ll \mu }$ .

The two measures are called equivalent iff ${\displaystyle \mu \ll \nu }$  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, ${\displaystyle {\mathcal {N}}_{\mu }=\{\emptyset \}}$ . So by the second definition, any other measure ${\displaystyle \nu }$  is equivalent to the counting measure iff 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.