Equivalence (measure theory)

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 $$\mathcal{N}_\mu := \{A \in \mathcal{A} \mid \mu(A) = 0\}$$ and $$\mathcal{N}_\nu := \{A \in \mathcal{A} \mid \nu(A) = 0\}$$ 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 $$\mathcal N_\mu = \mathcal N_\nu.$$

On the real line
Define the two measures on the real line as $$\mu(A)= \int_A \mathbf 1_{[0,1]}(x) \mathrm dx$$ $$\nu(A)= \int_A x^2 \mathbf 1_{[0,1]}(x) \mathrm dx$$ for all Borel sets $$A.$$ Then $$\mu$$ and $$\nu$$ are equivalent, since all sets outside of $$[0,1]$$ have $$\mu$$ and $$\nu$$ measure zero, and a set inside $$[0,1]$$ 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, $$\mathcal N_\mu = \{\varnothing\}.$$ 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 $$\mu.$$