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 $(X,{\mathcal {A}})$ , and let

{\mathcal {N}}_{\mu }:=\{A\in {\mathcal {A}}\mid \mu (A)=0\}

and

{\mathcal {N}}_{\nu }:=\{A\in {\mathcal {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 $\mu$ iff ${\mathcal {N}}_{\nu }\supseteq {\mathcal {N}}_{\mu }$ . This is denoted as $\nu \ll \mu$ .

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

Examples

On the real line

Define the two measures on the real line as

\mu (A)=\int _{A}\mathbf {1} _{[0,1]}(x)\mathrm {d} x

\nu (A)=\int _{A}x^{2}\mathbf {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 $(X,{\mathcal {A}})$ 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, ${\mathcal {N}}_{\mu }=\{\emptyset \}$ . So by the second definition, any other measure $\nu$ is equivalent to the counting measure iff it also has just the empty set as the only $\nu$ -null set.

Supporting measures

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

References

Klenke, Achim (2008). Probability Theory. Berlin: Springer. p. 156. doi:10.1007/978-1-84800-048-3. ISBN 978-1-84800-047-6.
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.

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[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.