Equivalence (measure theory)
Let and be two measures on the measurable space , and let
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
So by the second definition, any other measure is equivalent to the counting measure iff it also has just the empty set as the only null set.
A measure is called a supporting measure of a measure if is -finite and is equivalent to .