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IX: 23, 443-463, LNM 465 (1975)

**GETOOR, Ronald K.**

On the construction of kernels (Measure theory)

Given two measurable spaces $(E, {\cal E})$ $(F, {\cal F})$ and a family ${\cal N}\subset{\cal E}$ of negligible sets, a pseudo-kernel $T$ is a mapping from bounded measurable functions on $F$ to classes mod.${\cal N}$ of bounded measurable functions on $E$, which has all a.e. the properties (positivity, countable additivity) of a kernel. Regularizing $T$ consists in finding a true kernel $\hat T$ such that $\hat Tf$ belongs to the class $Tf$ for every measurable bounded $f$ on $F$. The regularization is easy whenever $F$ is compact metric. Then the result is extended to the case of a Lusin space, and to the case of a U-space (Radon space) assuming ${\cal N}$ consists of the negligible sets for a family of measures on $E$. An application is given to densities of continuous additive functionals of a Markov process

Comment: The author states that his paper is purely expository. This is not true, though the proof is a standard one in the theory of conditional distributions. For a deeper result, see Dellacherie 1030. For a presentation in book form, see Dellacherie-Meyer,*Probabilités et Potentiel C,* chapter XI **41**

Keywords: Pseudo-kernels, Regularization

Nature: Original

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IX: 24, 464-465, LNM 465 (1975)

**MEYER, Paul-André**

Une remarque sur la construction de noyaux (Measure theory)

With the notation of the preceding report 923, this is a first attempt to solve the case (important in practice) where $F$ is coanalytic, assuming ${\cal N}$ consists of the negligible sets of a Choquet capacity

Comment: See Dellacherie 1030

Keywords: Pseudo-kernels, Regularization

Nature: Original

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X: 30, 545-577, LNM 511 (1976)

**DELLACHERIE, Claude**

Sur la construction de noyaux boréliens (Measure theory)

This answers questions of Getoor 923 and Meyer 924 on the regularization of a pseudo-kernel relative to a family ${\cal N}$ of negligible sets into a Borel kernel. The problem is reduced to a simpler one, whether a non-negligible set $A$ contains a non-negligible Borel set, which itself is answered in the affirmative if 1) The underlying space is compact metric, 2) $A$ is coanalytic, 3) ${\cal N}$ consists of all sets negligible for all measures of an analytic family. The proof uses general methods, of independent interest

Comment: For a presentation in book form, see Dellacherie-Meyer,*Probabilités et Potentiel C,* chapter XI **41**. The hypothesis that the space is compact is sometimes troublesome for the applications

Keywords: Pseudo-kernels, Regularization

Nature: Original

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XI: 15, 298-302, LNM 581 (1977)

**ZANZOTTO, Pio Andrea**

Sur l'existence d'un noyau induisant un opérateur sous markovien donné (Measure theory)

The problem is whether a positive, norm-decreasing operator $L^\infty(\mu)\rightarrow L^\infty(\lambda)$ (of classes, not functions) is induced by a submarkov kernel. No ``countable additivity'' condition is assumed, but completeness of $\lambda$ and tightness of $\mu$

Comment: See 923, 924, 1030

Keywords: Pseudo-kernels, Regularization

Nature: Original

Retrieve article from Numdam

On the construction of kernels (Measure theory)

Given two measurable spaces $(E, {\cal E})$ $(F, {\cal F})$ and a family ${\cal N}\subset{\cal E}$ of negligible sets, a pseudo-kernel $T$ is a mapping from bounded measurable functions on $F$ to classes mod.${\cal N}$ of bounded measurable functions on $E$, which has all a.e. the properties (positivity, countable additivity) of a kernel. Regularizing $T$ consists in finding a true kernel $\hat T$ such that $\hat Tf$ belongs to the class $Tf$ for every measurable bounded $f$ on $F$. The regularization is easy whenever $F$ is compact metric. Then the result is extended to the case of a Lusin space, and to the case of a U-space (Radon space) assuming ${\cal N}$ consists of the negligible sets for a family of measures on $E$. An application is given to densities of continuous additive functionals of a Markov process

Comment: The author states that his paper is purely expository. This is not true, though the proof is a standard one in the theory of conditional distributions. For a deeper result, see Dellacherie 1030. For a presentation in book form, see Dellacherie-Meyer,

Keywords: Pseudo-kernels, Regularization

Nature: Original

Retrieve article from Numdam

IX: 24, 464-465, LNM 465 (1975)

Une remarque sur la construction de noyaux (Measure theory)

With the notation of the preceding report 923, this is a first attempt to solve the case (important in practice) where $F$ is coanalytic, assuming ${\cal N}$ consists of the negligible sets of a Choquet capacity

Comment: See Dellacherie 1030

Keywords: Pseudo-kernels, Regularization

Nature: Original

Retrieve article from Numdam

X: 30, 545-577, LNM 511 (1976)

Sur la construction de noyaux boréliens (Measure theory)

This answers questions of Getoor 923 and Meyer 924 on the regularization of a pseudo-kernel relative to a family ${\cal N}$ of negligible sets into a Borel kernel. The problem is reduced to a simpler one, whether a non-negligible set $A$ contains a non-negligible Borel set, which itself is answered in the affirmative if 1) The underlying space is compact metric, 2) $A$ is coanalytic, 3) ${\cal N}$ consists of all sets negligible for all measures of an analytic family. The proof uses general methods, of independent interest

Comment: For a presentation in book form, see Dellacherie-Meyer,

Keywords: Pseudo-kernels, Regularization

Nature: Original

Retrieve article from Numdam

XI: 15, 298-302, LNM 581 (1977)

Sur l'existence d'un noyau induisant un opérateur sous markovien donné (Measure theory)

The problem is whether a positive, norm-decreasing operator $L^\infty(\mu)\rightarrow L^\infty(\lambda)$ (of classes, not functions) is induced by a submarkov kernel. No ``countable additivity'' condition is assumed, but completeness of $\lambda$ and tightness of $\mu$

Comment: See 923, 924, 1030

Keywords: Pseudo-kernels, Regularization

Nature: Original

Retrieve article from Numdam