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

The two measures are called equivalent if and only if [math]\displaystyle{ \mu \ll \nu }[/math] and [math]\displaystyle{ \nu \ll \mu, }[/math]^[1] which is denoted as [math]\displaystyle{ \mu \sim \nu. }[/math] That is, two measures are equivalent if they satisfy [math]\displaystyle{ \mathcal N_\mu = \mathcal N_\nu. }[/math]

Examples

On the real line

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

Abstract measure space

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

Supporting measures

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

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