Equivalence (measure theory) explained

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

\mu and \nu be two measures on the measurable space (X, l A), and let $$\backslash mathcal\_\backslash mu\; :=\; \backslash$$and$$\backslash mathcal\_\backslash nu\; :=\; \backslash$$be the sets of \mu -null sets and \nu -null sets, respectively. Then the measure \nu is said to be absolutely continuous in reference to \mu if and only if l N _{\nu} \supseteq l N _{\mu.} This is denoted as \nu \ll \mu.

The two measures are called equivalent if and only if

\mu \ll \nu and \nu \ll \mu, ^[1] which is denoted as \mu \sim \nu. That is, two measures are equivalent if they satisfy l N _{\mu} = l N _{\nu.}

Examples

On the real line

Define the two measures on the real line as$$\backslash mu(A)=\; \backslash int\_A\; \backslash mathbf\; 1\_(x)\; \backslash mathrm\; dx$$$$\backslash nu(A)=\; \backslash int\_A\; x^2\; \backslash mathbf\; 1\_(x)\; \backslash mathrm\; dx$$for all Borel sets

A. Then \mu and \nu are equivalent, since all sets outside of [ 0 , 1 ] have \mu and \nu measure zero, and a set inside [ 0 , 1 ] is a \mu -null set or a \nu -null set exactly when it is a null set with respect to Lebesgue measure.

Abstract measure space

Look at some measurable space

(X, l A) and let \mu be the counting measure, so$$\backslash mu(A)\; =\; |A|,$$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, l N _{\mu} = \{\varnothing\}. So by the second definition, any other measure \nu is equivalent to the counting measure if and only if it also has just the empty set as the only \nu -null set.

Supporting measures

A measure

\mu is called a of a measure \nu if \mu is \sigma -finite and \nu is equivalent to \mu. ^[2]

Notes and References

1. Book: Klenke . Achim . 2008 . Probability Theory . Berlin . Springer . 10.1007/978-1-84800-048-3 . 978-1-84800-047-6. 156.
2. Book: Kallenberg . Olav . Olav Kallenberg . 2017 . Random Measures, Theory and Applications. Switzerland . Springer . 10.1007/978-3-319-41598-7. 978-3-319-41596-3. 21.