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.


Let   and   be two measures on the measurable space  , and let




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  .


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]


  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.