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
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 edit
On the real line edit
Define the two measures on the real line as
Abstract measure space edit
Look at some measurable space and let be the counting measure, so
Supporting measures edit
A 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.