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.
μ
be two measures on the measurable space
μ
∣ μ (
μ
-null sets, respectively.
is said to be absolutely continuous in reference to
μ
μ
ν ≪ μ .
The two measures are called equivalent if and only if
μ ≪ ν
ν ≪ μ ,
μ ∼ ν .
That is, two measures are equivalent if they satisfy
μ
Define the two measures on the real line as
for all Borel sets
are equivalent, since all sets outside of
measure zero, and a set inside
-null set exactly when it is a null set with respect to Lebesgue measure.
Look at some measurable space
be the counting measure, so
is the cardinality of the set a.
So the counting measure has only one null set, which is the empty set.
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
is called a supporting measure of a measure