概率空间乘积中提升的分裂
Splitting of liftings in products of probability spaces
摘要：
证明了：当（X，\mathfrakA，P）是具有可计数生成的σ-代数\mathfrakA的任意概率空间，（Y，\mathfrakB，Q）是具有提升ρ的任意完全概率空间，并且\hat R是\mathfrakA \hat \otimes_R \mathfrakB上的完全概率测度，由正则条件概率{S_y：y \in Y}在\mathfrakA上关于\mathfrakB，则存在提升π在（X \times Y，\mathfrakA \hat \otimes_R \mathfrakB，\hat R）和提升σ_y，（X，\hat \mathfrakA_y，\hat S_y），y \in Y，使得对于每个E \in \mathfrakA \hat \otimes_R \mathfrakB和每个y \in Y，[π（E）]^y=σ_yPDMATHABMT（[π（E）]^yPDMATHABMU）.假设R关于P \otimes Q是绝对连续的，我们证明了正则条件概率{T_y：y \in Y}和提升\varpi（X \times Y，\mathfrakA \hat \otimes_R \mathfrakB，\hat R），ρ' on（Y，\mathfrakB，\hat Q）和σ_y（X，\hat \mathfrakA_y，\hat S_y），y \in Y，使得对于每个E \in \mathfrakA \hat \otimes_R \mathfrakB和每个y \in Y，[ \varpi（E）]^y=σ_y \bigl（[ \varpi（E）]^y \bigr）和\varpi（A \times B）= \bigcup_{y\inρ'(B)} σ_y（A）\times{y} \qquadif A \times B \in \mathfrakA \times \mathfrakB .这两个结果都是Musiaeville，Strauss和Macheras [Fund.Math.166（2000）281-303]在测度不一定是边际测度的乘积的情况下的推广。我们还证明了本文所得到的提升总是将\hat R-可测随机过程转化为它们的\hat R-可测修改。
英文摘要：
We prove that if (X,\mathfrakA,P) is an arbitrary probability space with countably generated σ-algebra \mathfrakA, (Y,\mathfrakB,Q) is an arbitrary complete probability space with a lifting ρand \hat R is a complete probability measure on \mathfrakA \hat \otimes_R \mathfrakB determined by a regular conditional probability {S_y:y\in Y} on \mathfrakA with respect to \mathfrakB, then there exist a lifting πon (X\times Y,\mathfrakA \hat \otimes_R \mathfrakB,\hat R) and liftings σ_y on (X,\hat \mathfrakA_y,\hat S_y), y\in Y, such that, for every E\in\mathfrakA \hat \otimes_R \mathfrakB and every y\in Y, [π(E)]^y=σ_y\bigl([π(E)]^y\bigr). Assuming the absolute continuity of R with respect to P\otimes Q, we prove the existence of a regular conditional probability {T_y:y\in Y} and liftings \varpi on (X\times Y,\mathfrakA \hat \otimes_R \mathfrakB,\hat R), ρ' on (Y,\mathfrakB,\hat Q) and σ_y on (X,\hat \mathfrakA_y,\hat S_y), y\in Y, such that, for every E\in\mathfrakA \hat \otimes_R \mathfrakB and every y\in Y, [\varpi(E)]^y=σ_y\bigl([\varpi(E)]^y\bigr) and \varpi(A\times B)=\bigcup_{y\inρ'(B)}σ_y(A)\times{y}\qquadif A\times B\in\mathfrakA\times\mathfrakB. 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 \hat R-measurable stochastic processes into their \hat R-measurable modifications.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
