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.

## Contents

* 1 Definition
* 2 Examples
* 2.1 On the real line
* 2.2 Abstract measure space
* 3 Supporting measures
* 4 References

## Definition

Let μ and ν be two measures on the measurable space (X,𝒜), and let 𝒩μ:={A∈𝒜∣μ(A)=0} and 𝒩ν:={A∈𝒜∣ν(A)=0} be the sets of μ-null sets and ν-null sets, respectively. Then the measure ν is said to be absolutely continuous in reference to μ if and only if 𝒩ν⊇𝒩μ. This is denoted as ν≪μ.

The two measures are called equivalent if and only if μ≪ν and ν≪μ,[1] which is denoted as μ∼ν. That is, two measures are equivalent if they satisfy 𝒩μ=𝒩ν.

## Examples

### On the real line

Define the two measures on the real line as μ(A)=∫A𝟏[0,1](x)dx ν(A)=∫Ax2𝟏[0,1](x)dx for all Borel sets A. Then μ and ν are equivalent, since all sets outside of [0,1] have μ and ν measure zero, and a set inside [0,1] is a μ-null set or a ν-null set exactly when it is a null set with respect to Lebesgue measure.

### Abstract measure space

Look at some measurable space (X,𝒜) and let μ be the counting measure, so μ(A)=|A|, where |A| is the cardinality of the set a. So the counting measure has only one null set, which is the empty set. That is, 𝒩μ={∅}. So by the second definition, any other measure ν is equivalent to the counting measure if and only if it also has just the empty set as the only ν-null set.

## Supporting measures

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

## References

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.

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