Equivalence Of Measures

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 $(X, \mathcal A),$ and let
$\mathcal_\mu := \$
and
$\mathcal_\nu := \$
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 $\mathcal N_\nu \supseteq \mathcal N_\mu.$ This is denoted as $\nu \ll \mu.$

The two measures are called equivalent if and only if $\mu \ll \nu$ and $\nu \ll \mu,$ which is denoted as $\mu \sim \nu.$ That is, two measures are equivalent if they satisfy $\mathcal N_\mu = \mathcal N_\nu.$

Examples

On the real line

Define the two measures on the real line as
$\mu(A)= \int_A \mathbf 1_(x) \mathrm dx$
$\nu(A)= \int_A x^2 \mathbf 1_(x) \mathrm dx$
for all Borel sets $A.$ Then $\mu$ and $\nu$ are equivalent, since all sets outside of ,1/math> have $\mu$ and $\nu$ measure zero, and a set inside ,1/math> 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 $(X, \mathcal A)$ and let $\mu$ be the counting measure, so
$\mu(A) = , A, ,$
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, $\mathcal N_\mu = \.$ 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 $\mu.$

References

{{Measure theory

Equivalence (mathematics)
Measure theory