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July 2004 Splitting of liftings in products of probability spaces
N. D. Macheras, K. Musiał, W. Strauss
Ann. Probab. 32(3B): 2389-2408 (July 2004). DOI: 10.1214/009117904000000018


We prove that if $(X,{\mathfrak{A}},P)$ is an arbitrary probability space with countably generated σ-algebra ${\mathfrak{A}}$, $(Y,{\mathfrak{B}},Q)$ is an arbitrary complete probability space with a lifting ρ and $\widehat {R}$ is a complete probability measure on ${\mathfrak{A}}{\,\widehat{\otimes}_{R}\,}{\mathfrak{B}}$ determined by a regular conditional probability $\{S_y:y∈Y\}$ on ${\mathfrak{A}}$ with respect to ${\mathfrak{B}}$, then there exist a lifting π on $(X\times Y,{\mathfrak{A}}{\,\widehat{\otimes}_{R}\,}{\mathfrak{B}},\widehat {R})$ and liftings $σ_y$ on $(X,\widehat {\mathfrak{A}}_{y},\widehat {S}_{y})$, $y∈Y$, such that, for every $E\in{\mathfrak{A}}{\,\widehat{\otimes}_{R}\,}{\mathfrak{B}}$ and every $y∈Y$, $$[\pi(E)]^{y}=\sigma_{y}\bigl([\pi(E)]^{y}\bigr).$$ Assuming the absolute continuity of $R$ with respect to $P⊗Q$, we prove the existence of a regular conditional probability $\{T_y:y∈Y\}$ and liftings ϖ on $(X\times Y,{\mathfrak{A}}{\,\widehat{\otimes}_{R}\,}{\mathfrak{B}},\widehat {R})$, ρ' on $(Y,\mathfrak{B},\widehat {Q})$ and $σ_y$ on $(X,\widehat {\mathfrak{A}}_{y},\widehat {S}_{y})$, $y∈Y$, such that, for every $E\in{\mathfrak{A}}{\,\widehat{\otimes}_{R}\,}{\mathfrak{B}}$ and every $y∈Y$, $$[\varpi(E)]^{y}=\sigma_{y}\bigl([\varpi(E)]^{y}\bigr)$$ and $$\varpi(A\times B)=\bigcup_{y\in\rho'(B)}\sigma_{y}(A)\times\{y\}\qquad\mbox{if }A\times B\in{\mathfrak{A}}\times{\mathfrak{B}}.$$ Both results are generalizations of Musiał, Strauss and Macheras [Fund. Math. 166 (2000) 281–303] to the case of measures which are not necessarily products of marginal measures. We prove also that liftings obtained in this paper always convert $\widehat {R}$-measurable stochastic processes into their $\widehat {R}$-measurable modifications.


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N. D. Macheras. K. Musiał. W. Strauss. "Splitting of liftings in products of probability spaces." Ann. Probab. 32 (3B) 2389 - 2408, July 2004.


Published: July 2004
First available in Project Euclid: 6 August 2004

zbMATH: 1058.60005
MathSciNet: MR2078544
Digital Object Identifier: 10.1214/009117904000000018

Primary: 28A35 , 28A50 , 28A51 , 60A01 , 60G05

Keywords: densities , Liftings , measurable stochastic processes , product densities , product liftings , product measures , regular conditional probabilities

Rights: Copyright © 2004 Institute of Mathematical Statistics


Vol.32 • No. 3B • July 2004
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