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
editLet and be two measures on the measurable space and let
The two measures are called equivalent if and only if and [1] which is denoted as That is, two measures are equivalent if they satisfy
Examples
editOn the real line
editDefine the two measures on the real line as
Abstract measure space
editLook at some measurable space and let be the counting measure, so
Supporting measures
editA measure is called a supporting measure of a measure if is -finite and is equivalent to [2]
References
edit- ^ Klenke, Achim (2008). Probability Theory. Berlin: Springer. p. 156. doi:10.1007/978-1-84800-048-3. ISBN 978-1-84800-047-6.
- ^ 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.