Equivalent 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 $\mu$ iff $\mathcal N_\nu \supseteq \mathcal N_\mu$. This is denoted as $\nu \ll \mu$.

The two measures are called equivalent iff $\mu \ll \nu$ 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 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 iff it also has just the empty set as the only $\nu$-null set.

Supporting measures

A measure $\mu$ is called a supporting measure of a measure $\nu$ if $\mu$ is $\sigma$-finite and $\nu$ is equivalent to $\mu$.

References

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Measure theory
Equivalence (mathematics)