The Annals of Probability

Splitting of liftings in products of probability spaces

N. D. Macheras, K. Musiał, and W. Strauss

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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 {Sy:yY} 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 σy on $(X,\widehat {\mathfrak{A}}_{y},\widehat {S}_{y})$, yY, such that, for every $E\in{\mathfrak{A}}{\,\widehat{\otimes}_{R}\,}{\mathfrak{B}}$ and every yY, \[[\pi(E)]^{y}=\sigma_{y}\bigl([\pi(E)]^{y}\bigr).\] Assuming the absolute continuity of R with respect to PQ, we prove the existence of a regular conditional probability {Ty:yY} and liftings ϖ on $(X\times Y,{\mathfrak{A}}{\,\widehat{\otimes}_{R}\,}{\mathfrak{B}},\widehat {R})$, ρ' on $(Y,\mathfrak{B},\widehat {Q})$ and σy on $(X,\widehat {\mathfrak{A}}_{y},\widehat {S}_{y})$, yY, such that, for every $E\in{\mathfrak{A}}{\,\widehat{\otimes}_{R}\,}{\mathfrak{B}}$ and every yY, \[[\varpi(E)]^{y}=\sigma_{y}\bigl([\varpi(E)]^{y}\bigr)\] 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.

Article information

Ann. Probab., Volume 32, Number 3B (2004), 2389-2408.

First available in Project Euclid: 6 August 2004

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Zentralblatt MATH identifier

Primary: 28A51: Lifting theory [See also 46G15] 28A50: Integration and disintegration of measures 28A35: Measures and integrals in product spaces 60A01 60G05: Foundations of stochastic processes

Liftings product liftings product measures regular conditional probabilities densities product densities measurable stochastic processes


Strauss, W.; Macheras, N. D.; Musiał, K. Splitting of liftings in products of probability spaces. Ann. Probab. 32 (2004), no. 3B, 2389--2408. doi:10.1214/009117904000000018.

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