Equivalence (measure theory)

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. ==Examples==

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. ==References==