Equivalence (measure theory)

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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

be the set of all -null sets; is similarly defined. Then the measure is said to be absolutely continuous in reference to iff . This is denoted as .

The two measures are called equivalent iff and ,[1] which is denoted as . An equivalent definition is that two measures are equivalent if they satisfy .


On the real line[edit]

Define the two measures on the real line as

for all Borel sets . Then and are equivalent, since all sets outside of have measure zero, and a set inside is a null set in respect to the Lebesgue measure.

Abstract measure space[edit]

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. Therefore,


So by the second definition, any other measure is equivalent to the counting measure iff it also has just the empty set as the only null set.

Supporting measures[edit]

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


  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.