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.

Definition[edit]

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 .

Examples[edit]

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]

References[edit]

  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.