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

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Let   and   be two measures on the measurable space   and let   and   be the sets of  -null sets and  -null sets, respectively. Then the measure   is said to be absolutely continuous in reference to   if and only if   This is denoted as  

The two measures are called equivalent if and only if   and  [1] which is denoted as   That is, two measures are equivalent if they satisfy  

Examples

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On the real line

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Define the two measures on the real line as     for all Borel sets   Then   and   are equivalent, since all sets outside of   have   and   measure zero, and a set inside   is a  -null set or a  -null set exactly when it is a null set with respect to Lebesgue measure.

Abstract measure space

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Look at some measurable space   and let   be the counting measure, so   where   is the cardinality of the set a. So the counting measure has only one null set, which is the empty set. That is,   So by the second definition, any other measure   is equivalent to the counting measure if and only if it also has just the empty set as the only  -null set.

Supporting measures

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A measure   is called a supporting measure of a measure   if   is  -finite and   is equivalent to  [2]

References

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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.