Equivalence (measure theory)

Let $\backslash mu$ and $\backslash nu$ be two measures on the measurable space $(X,\; \backslash mathcal\; A),$ and let

$$\backslash mathcal\{N\}\_\backslash mu\; :=\; \backslash \{A\; \backslash in\; \backslash mathcal\{A\}\; \backslash mid\; \backslash mu(A)\; =\; 0\backslash \}$$

and

$$\backslash mathcal\{N\}\_\backslash nu\; :=\; \backslash \{A\; \backslash in\; \backslash mathcal\{A\}\; \backslash mid\; \backslash nu(A)\; =\; 0\backslash \}$$

be the sets of $\backslash mu$-null sets and $\backslash nu$-null sets, respectively. Then the measure $\backslash nu$ is said to be absolutely continuous in reference to $\backslash mu$ if and only if $\backslash mathcal\; N\_\backslash nu\; \backslash supseteq\; \backslash mathcal\; N\_\backslash mu.$ This is denoted as $\backslash nu\; \backslash ll\; \backslash mu.$

The two measures are called equivalent if and only if $\backslash mu\; \backslash ll\; \backslash nu$ and $\backslash nu\; \backslash ll\; \backslash mu,${{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}} which is denoted as $\backslash mu\; \backslash sim\; \backslash nu.$ That is, two measures are equivalent if they satisfy $\backslash mathcal\; N\_\backslash mu\; =\; \backslash mathcal\; N\_\backslash nu.$

Define the two measures on the real line as

$$\backslash mu(A)=\; \backslash int\_A\; \backslash mathbf\; 1\_\{[0,1]\}(x)\; \backslash mathrm\; dx$$

$$\backslash nu(A)=\; \backslash int\_A\; x^2\; \backslash mathbf\; 1\_\{[0,1]\}(x)\; \backslash mathrm\; dx$$

for all Borel sets $A.$ Then $\backslash mu$ and $\backslash nu$ are equivalent, since all sets outside of $[0,1]$ have $\backslash mu$ and $\backslash nu$ measure zero, and a set inside $[0,1]$ is a $\backslash mu$-null set or a $\backslash nu$-null set exactly when it is a null set with respect to Lebesgue measure.

Look at some measurable space $(X,\; \backslash mathcal\; A)$ and let $\backslash mu$ be the counting measure, so

$$\backslash mu(A)\; =\; |A|,$$

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, $\backslash mathcal\; N\_\backslash mu\; =\; \backslash \{\backslash varnothing\backslash \}.$ So by the second definition, any other measure $\backslash nu$ is equivalent to the counting measure if and only if it also has just the empty set as the only $\backslash nu$-null set.

A measure $\backslash mu$ is called a {{visible anchor|supporting measure}} of a measure $\backslash nu$ if $\backslash mu$ is $\backslash sigma$-finite and $\backslash nu$ is equivalent to $\backslash mu.${{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}}

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Category:Equivalence (mathematics)

Category:Measure theory