View source for Equivalence (measure theory)

In [[mathematics]], and specifically in [[measure theory]], '''equivalence''' is a notion of two [[Measure (mathematics)|measures]] being qualitatively similar. Specifically, the two measures agree on which events have measure zero.

==Definition==

Let <math>\mu</math> and <math>\nu</math> be two [[Measure (mathematics)|measures]] on the measurable space <math>(X, \mathcal A),</math> and let 
<math display=block>\mathcal{N}_\mu := \{A \in \mathcal{A} \mid \mu(A) = 0\}</math>
and
<math display=block>\mathcal{N}_\nu := \{A \in \mathcal{A} \mid \nu(A) = 0\}</math>
be the sets of <math>\mu</math>-[[null set]]s and <math>\nu</math>-null sets, respectively. Then the measure <math>\nu</math> is said to be [[Absolutely continuous measure|absolutely continuous]] in reference to <math>\mu</math> if and only if <math>\mathcal N_\nu \supseteq \mathcal N_\mu.</math> This is denoted as <math>\nu \ll \mu.</math>

The two measures are called equivalent if and only if <math>\mu \ll \nu</math> and <math>\nu \ll \mu,</math><ref>{{cite book|last1=Klenke |first1=Achim |year=2008  |title=Probability Theory |location=Berlin |publisher=Springer |doi=10.1007/978-1-84800-048-3 |isbn=978-1-84800-047-6|page=156}}</ref> which is denoted as <math>\mu \sim \nu.</math> That is, two measures are equivalent if they satisfy <math>\mathcal N_\mu = \mathcal N_\nu.</math>

==Examples==

===On the real line===

Define the two measures on the [[real line]] as
<math display=block>\mu(A)= \int_A \mathbf 1_{[0,1]}(x) \mathrm dx</math>
<math display=block>\nu(A)= \int_A x^2 \mathbf 1_{[0,1]}(x) \mathrm dx</math>
for all [[Borel set]]s <math>A.</math> Then <math>\mu</math> and <math>\nu</math> are equivalent, since all sets outside of <math>[0,1]</math> have <math>\mu</math> and <math>\nu</math> measure zero, and a set inside <math>[0,1]</math> is a <math>\mu</math>-null set or a <math>\nu</math>-null set exactly when it is a null set with respect to [[Lebesgue measure]].

===Abstract measure space===

Look at some measurable space <math>(X, \mathcal A)</math> and let <math>\mu</math> be the [[counting measure]], so
<math display=block>\mu(A) = |A|,</math>
where <math>|A|</math> is the [[cardinality]] of the set a. So the counting measure has only one null set, which is the [[empty set]]. That is, <math>\mathcal N_\mu = \{\varnothing\}.</math> So by the second definition, any other measure <math>\nu</math> is equivalent to the counting measure if and only if it also has just the empty set as the only <math>\nu</math>-null set.

==Supporting measures==

A measure <math>\mu</math> is called a '''{{visible anchor|supporting measure}}''' of a measure <math>\nu</math> if <math>\mu</math> is [[sigma-finite|<math>\sigma</math>-finite]] and <math>\nu</math> is equivalent to <math>\mu.</math><ref>{{cite book |last1=Kallenberg |first1=Olav |author-link1=Olav Kallenberg |year=2017  |title=Random Measures, Theory and Applications|location= Switzerland |publisher=Springer |doi= 10.1007/978-3-319-41598-7|isbn=978-3-319-41596-3|page=21}}</ref>

==References==

{{reflist}}

{{Measure theory}}

[[Category:Equivalence (mathematics)]]
[[Category:Measure theory]]