##
**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 \({\mathcal T}\).

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 |