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.

DefinitionEdit

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   iff  . This is denoted as  .

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

ExamplesEdit

On the real lineEdit

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 spaceEdit

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 iff it also has just the empty set as the only  -null set.

Supporting measuresEdit

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

ReferencesEdit

  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.