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| In [[probability theory]], a '''random measure''' is a [[measure (mathematics)|measure]]-valued [[random element]].<ref name="RN">[[Olav Kallenberg|Kallenberg, O.]], ''Random Measures'', 4th edition. Academic Press, New York, London; Akademie-Verlag, Berlin (1986). ISBN 0-12-394960-2 {{MR|854102}}. An authoritative but rather difficult reference.
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| </ref><ref name="G">Jan Grandell, Point processes and random measures, ''Advances in Applied Probability'' 9 (1977) 502-526. {{MR|0478331}} [http://links.jstor.org/sici?sici=0001-8678%28197709%299%3A3%3C502%3APPARM%3E2.0.CO%3B2-5 JSTOR] A nice and clear introduction.
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| </ref> Let X be a complete separable metric space and <math>\mathfrak{B}(X)</math> the [[Sigma-algebra#Generated .CF.83-algebra|σ-algebra]] of its Borel sets. A [[Borel measure]] μ on X is boundedly finite if μ(A) < ∞ for every bounded Borel set A. Let <math>M_X</math> be the space of all boundedly finite measures on <math>\mathfrak{B}(X)</math>. Let {{nowrap|(Ω, ℱ, ''P'')}} be a [[probability space]], then a random measure maps from this probability space to the [[measurable space]] {{nowrap|(<math>M_X</math>, <math>\mathfrak{B}(M_X)</math>)}}.<ref name="daleyPPI2003">{{cite doi|10.1007/b97277}}</ref>A measure generally might be decomposed as:
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| :<math> \mu=\mu_d + \mu_a = \mu_d + \sum_{n=1}^N \kappa_n \delta_{X_n}, </math>
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| Here <math>\mu_d</math> is a diffuse measure without atoms, while <math>\mu_a</math> is a purely atomic measure.
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| ==Random counting measure==
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| A random measure of the form:
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| :<math> \mu=\sum_{n=1}^N \delta_{X_n}, </math>
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| where <math>\delta</math> is the [[Dirac measure]], and <math>X_n</math> are random variables, is called a ''[[point process]]''<ref name="RN"/><ref name="G"/> or [[random counting measure]]. This random measure describes the set of ''N'' particles, whose locations are given by the (generally vector valued) random variables <math>X_n</math>. The diffuse component <math>\mu_d</math> is null for a counting measure.
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| In the formal notation of above a random counting measure is a map from a probability space to the measurable space {{nowrap|(<math>N_X</math>, <math>\mathfrak{B}(N_X)</math>)}} a [[measurable space]]. Here <math>N_X</math> is the space of all boundedly finite integer-valued measures <math>N \in M_X</math> (called counting measures).
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| The definitions of expectation measure, Laplace functional, moment measures and stationarity for random measures follow those of [[point process]]es. Random measures are useful in the description and analysis of [[Monte Carlo method]]s, such as [[numerical quadrature#Monte Carlo|Monte Carlo numerical quadrature]] and [[particle filter]]s.<ref>Crisan, D., ''Particle Filters: A Theoretical Perspective'', in ''Sequential Monte Carlo in Practice,'' Doucet, A., de Freitas, N. and Gordon, N. (Eds), Springer, 2001, ISBN 0-387-95146-6</ref>
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| ==See also==
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| * [[Point process]]
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| * [[Poisson random measure]]
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| * [[Random element]]
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| * [[Vector measure]]
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| * [[Distribution ensemble|Ensemble]]
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| ==References==
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| <references/>
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| [[Category:Measures (measure theory)]]
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| [[Category:Stochastic processes]]
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| {{probability-stub}}
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