Existence of measurable versions of stochastic processes
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866912948145881088 |
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| author | Musiał, Kazimierz |
| author_facet | Musiał, Kazimierz |
| contents | Let $(X, \mfA,P)$, $(Y, \mfB,Q)$ be two arbitrary probability spaces and $¶:=\{(\mfA,P_y):y\in{Y}\}$ be a regular conditional probability on $\mfA$ with respect to $Q$. Denote by $R$ the skew product of $P$ and $Q$ determined by $\{P_y:y\in{Y}\}$ on the product $σ$-algebra $\mfA\otimes\mfB$ and by $\wh{R}$ its completion. I prove that a process $\{ξ_y:y\in{Y}\}$ possesses an equivalent $\wh{R}$-measurable version if and only if it is measurable with respect to a certain particular $σ$-algebra, larger than $\mfA\otimes\mfB$ and uniquely determined by $¶$. It is known that not every process possesses an equivalent measurable version (cf. \cite[§19.5]{St}).
My approach is essentially different from earlier trials. It reverts to \cite[Theorem 3]{ta1}, where Talagrand proved existence of an equivalent separable version of a measurable process (in case of $R=P\times{Q}$), provided $Y$ is endowed with a separable pseudometric. The theorem is a strong generalization of \cite[Theorem 6.1]{smm} and \cite[Theorem 5.1]{mms1} where it was proved only that a suitable class of liftings transfer a measurable process into a measurable process. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2603_06175 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Existence of measurable versions of stochastic processes Musiał, Kazimierz Probability 60G07, 28A35, 26B99 Let $(X, \mfA,P)$, $(Y, \mfB,Q)$ be two arbitrary probability spaces and $¶:=\{(\mfA,P_y):y\in{Y}\}$ be a regular conditional probability on $\mfA$ with respect to $Q$. Denote by $R$ the skew product of $P$ and $Q$ determined by $\{P_y:y\in{Y}\}$ on the product $σ$-algebra $\mfA\otimes\mfB$ and by $\wh{R}$ its completion. I prove that a process $\{ξ_y:y\in{Y}\}$ possesses an equivalent $\wh{R}$-measurable version if and only if it is measurable with respect to a certain particular $σ$-algebra, larger than $\mfA\otimes\mfB$ and uniquely determined by $¶$. It is known that not every process possesses an equivalent measurable version (cf. \cite[§19.5]{St}). My approach is essentially different from earlier trials. It reverts to \cite[Theorem 3]{ta1}, where Talagrand proved existence of an equivalent separable version of a measurable process (in case of $R=P\times{Q}$), provided $Y$ is endowed with a separable pseudometric. The theorem is a strong generalization of \cite[Theorem 6.1]{smm} and \cite[Theorem 5.1]{mms1} where it was proved only that a suitable class of liftings transfer a measurable process into a measurable process. |
| title | Existence of measurable versions of stochastic processes |
| topic | Probability 60G07, 28A35, 26B99 |
| url | https://arxiv.org/abs/2603.06175 |