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Concept# Equivalence (measure theory)

Summary

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.
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 which is denoted as That is, two measures are equivalent if they satisfy
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.
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.

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