# 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.

## Definition

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

### On the real line

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 space

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

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

## References

- 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.