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
and
be the sets of
-
null sets and
-null sets, respectively. Then the measure
is said to be
absolutely continuous
in reference to
if and only if
This is denoted as
The two measures are called equivalent if and only if and [1] which is denoted as That is, 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. That is,
So by the second definition, any other measure
is equivalent to the counting measure if and only if 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