## The existence of regular conditional probabilities: Necessary and sufficient conditions.(English)Zbl 0593.60002

Let (Y,$${\mathcal T},\mu)$$ be a probability space, (X,$${\mathcal S})$$ a measurable space and $$\nu: Y\times {\mathcal S}\to [0,1]$$ a kernel. A probability space (X,$${\mathcal S},\mu)$$ has the product regular conditional probability property (rcpp) if for any probability space (X$$\times Y,{\mathcal S}\times {\mathcal T},\lambda)$$ for which $$\mu$$ is the X-marginal of $$\lambda$$ there exists a kernel $$\nu: Y\times {\mathcal S}\to [0,1]$$ satisfying $\lambda (E\times F)=\int_{E}\nu (y,F)\lambda_ Y(dy)$ for all $$E\in {\mathcal T}$$, $$F\in {\mathcal S}$$, where $$\lambda_ Y$$ is the Y- marginal of $$\lambda$$.
Let (R,$${\mathcal B})$$ be the real line with Borel sets. (X,$${\mathcal S},\mu)$$ has the quotient rcpp if for every measurable $$f: X\to R$$ there exists a kernel $$\nu: R\times {\mathcal S}\to [0,1]$$ such that $\mu (F\cap f^{- 1}(E))=\int_{E}\nu (t,F)\mu_ R(dt)$ for al $$E\in {\mathcal B},\quad F\in {\mathcal S}$$, where $$\mu_ R$$ is the range of $$\mu$$ induced by f on $${\mathcal B}$$. If $$\nu$$ ($$\cdot,F)$$ is measurable only with respect to the completion of $${\mathcal B}$$ with respect to $$\mu_ R$$, then we say that (X,$${\mathcal S},\mu)$$ has D-quotient rcpp.
(X,$${\mathcal S},\mu)$$ has the subfield rcpp if for every sub-$$\sigma$$-field $${\mathcal T}\subset {\mathcal S}$$ there exists a kernel $$\nu: X\times {\mathcal S}\to [0,1]$$ (with $$\nu$$ ($$\cdot,F)$$ being $${\mathcal T}$$-measurable for all $$F\in {\mathcal S})$$ satisfying $\mu (F\cap E)=\int_{E}\nu (x,F)\mu (dx)$ for all $$F\in {\mathcal S},\quad E\in {\mathcal T}$$. Several relations between the above notions are examined in the paper. The following seem to be the most interesting:
1. If $${\mathcal S}$$ is countably generated then the perfectness of $$\mu$$, D- quotient rcpp, quotient rcpp, product rcpp, and the pre-standardness of (X,$${\mathcal S},\mu)$$ are equivalent.
2. If (X,$${\mathcal S},\mu)$$ is complete then the discreteness of $$\mu$$, product rcpp, subfield rcpp together with perfectness of $$\mu$$, and the quotient rcpp are equivalent. If one assumes the non-measurability of the continuum, then one may add to this list the D-quotient rcpp.
Remarks. a) The equivalence of the perfectness of $$\mu$$ to the product rcpp has been independently proved by W. Adamski [”Factorization of measures and perfection.” Proc. Am. Math. Soc. 97, 30-32 (1986)] who formulated it in a form suitable for arbitrary $${\mathcal S}$$. b) In the formulation of 2. $$(=Theorem$$ 11) the perfectness of $$\mu$$ is superfluous. This follows from the fact, that if (X,$${\mathcal S},\mu)$$ has subfield rcpp and $${\mathcal U}$$ is a sub-$$\sigma$$-field of $${\mathcal S}$$ then (X,$${\mathcal U},\mu)$$ also has the subfield rcpp. Taking an arbitrary countably generated $${\mathcal T}\subset {\mathcal S}$$ and its $$\mu$$ $$| {\mathcal T}$$-completion $${\mathcal T}^*$$ (we consider only sets being unions of $${\mathcal T}$$-atoms) and using the subfield rcpp of $${\mathcal T}^*$$ we can find $$T\in {\mathcal T}$$ of measure zero and such that $${\mathcal T}_{\cap}(X\setminus T)={\mathcal T}^*_{\cap}(X\setminus T).$$ This yields the discreteness of $$\mu$$ on $${\mathcal T}$$.
Reviewer: K.Musial

### MSC:

 60A10 Probabilistic measure theory 28A50 Integration and disintegration of measures 28A99 Classical measure theory
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