Equivalence (measure theory)

Last updated July 12, 2019

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.

Mathematics Field of study concerning quantity, patterns and change

Mathematics includes the study of such topics as quantity, structure (algebra), space (geometry), and change. It has no generally accepted definition.

In mathematical analysis, a measure on a set is a systematic way to assign a number to each suitable subset of that set, intuitively interpreted as its size. In this sense, a measure is a generalization of the concepts of length, area, and volume. A particularly important example is the Lebesgue measure on a Euclidean space, which assigns the conventional length, area, and volume of Euclidean geometry to suitable subsets of the $n$ -dimensional Euclidean space $R n$ . For instance, the Lebesgue measure of the interval $[0, 1]$ in the real numbers is its length in the everyday sense of the word, specifically, 1.

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\}

be the set of all $\mu$ -null sets; ${\mathcal {N}}_{\nu }$ is similarly defined. Then the measure $\nu$ is said to be absolutely continuous in reference to $\mu$ iff ${\mathcal {N}}_{\nu }\supset {\mathcal {N}}_{\mu }$ . This is denoted as $\nu \ll \mu$ .

In mathematical analysis, a null set $is a set that can be covered by a countable union of intervals of arbitrarily small total length. The notion of null set in set theory anticipates the development of Lebesgue measure since a null set necessarily has measure zero . More generally, on a given measure space a null set is a set such that .$

The two measures are called equivalent iff $\mu \ll \nu$ and $\nu \ll \mu$ ,^[1] which is denoted as $\mu \sim \nu$ . An equivalent definition is that 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 {d} x

\nu (A)=\int _{A}x^{2}\mathbf {1} _{[0,1]}(x)\mathrm {d} x

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.

In mathematics, a Borel set is any set in a topological space that can be formed from open sets through the operations of countable union, countable intersection, and relative complement. Borel sets are named after Émile Borel.

In measure theory, the Lebesgue measure, named after French mathematician Henri Lebesgue, is the standard way of assigning a measure to subsets of n-dimensional Euclidean space. For n = 1, 2, or 3, it coincides with the standard measure of length, area, or volume. In general, it is also called n-dimensional volume, n-volume, or simply volume. It is used throughout real analysis, in particular to define Lebesgue integration. Sets that can be assigned a Lebesgue measure are called Lebesgue-measurable; the measure of the Lebesgue-measurable set A is here denoted by λ(A).

Abstract measure space

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

In mathematics, the counting measure is an intuitive way to put a measure on any set: the "size" of a subset is taken to be: the number of elements in the subset if the subset has finitely many elements, and ∞ if the subset is infinite.

\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. Therefore,

{\mathcal {N}}_{\mu }=\{\emptyset \}

.

So by the second definition, any other measure $\nu$ is equivalent to the counting measure iff it also has just the empty set as the only null set.

Supporting measures

A measure $\mu$ is called a supporting measure of a measure $\nu$ if $\mu$ is $\sigma$ -finite and $\nu$ is equivalent to $\mu$ .^[2]

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References

↑ Klenke, Achim (2008). Probability Theory. Berlin: Springer. p. 156. doi:10.1007/978-1-84800-048-3. ISBN 978-1-84800-047-6.
↑ Kallenberg, Olav (2017). Random Measures, Theory and Applications. Switzerland: Springer. p. 21. doi:10.1007/978-3-319-41598-7. ISBN 978-3-319-41596-3.

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[1] Klenke, Achim (2008). Probability Theory. Berlin: Springer. p. 156. doi:10.1007/978-1-84800-048-3. ISBN 978-1-84800-047-6.

[2] Kallenberg, Olav (2017). Random Measures, Theory and Applications. Switzerland: Springer. p. 21. doi:10.1007/978-3-319-41598-7. ISBN 978-3-319-41596-3.