Proof that thin sets are finely separated
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I would like to justify the following claim concerning thin sets found in Chung, Lectures from Markov Processes to Brownian Motion, page 112:
Such a set is finely separated in the sense that each of its points has a fine neighborhood containing no other point of the set.
The context is the general theory of Hunt processes. The state space, which is assumed metrizable and compact, is denoted $mathcal{E}_{partial}$.
A (non-necessarily Borel) set $A$ is said to be finely open is for each $x in A$, there is a Borel set $B subset A$ such that $P^{x}[T_{B^c} > 0] = 1$ where $T_{E} equiv inf{ t > 0,;, X_t in E }$ denotes the hitting time of a (nearly-)Borel set $E$ for the Hunt process $(X_t)_{t geq 0}$ and $P^x$ is the probability measure with initial distribution concentrated on $x$. (Hence, if a set $A$ is finely open, then for all $x in A$, $P^x$-almost surely a sample path remains in $A$ for a finite amount of time.) In particular, an open set in $mathcal{E}_{partial}$ is finely open by right-continuity of the sample paths of the Hunt process.
A nearly-Borel set $A$ is said to be thin is for all $x in mathcal{E}_{partial}$, $P^x[T_B > 0] = 1$.
general-topology markov-process stopping-times
asked Dec 18 '18 at 10:25
IchKenneDeinenNamenIchKenneDeinenNamen
365
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I would like to justify the following claim concerning thin sets found in Chung, Lectures from Markov Processes to Brownian Motion, page 112:
Such a set is finely separated in the sense that each of its points has a fine neighborhood containing no other point of the set.
The context is the general theory of Hunt processes. The state space, which is assumed metrizable and compact, is denoted $mathcal{E}_{partial}$.
A (non-necessarily Borel) set $A$ is said to be finely open is for each $x in A$, there is a Borel set $B subset A$ such that $P^{x}[T_{B^c} > 0] = 1$ where $T_{E} equiv inf{ t > 0,;, X_t in E }$ denotes the hitting time of a (nearly-)Borel set $E$ for the Hunt process $(X_t)_{t geq 0}$ and $P^x$ is the probability measure with initial distribution concentrated on $x$. (Hence, if a set $A$ is finely open, then for all $x in A$, $P^x$-almost surely a sample path remains in $A$ for a finite amount of time.) In particular, an open set in $mathcal{E}_{partial}$ is finely open by right-continuity of the sample paths of the Hunt process.
A nearly-Borel set $A$ is said to be thin is for all $x in mathcal{E}_{partial}$, $P^x[T_B > 0] = 1$.
general-topology markov-process stopping-times
asked Dec 18 '18 at 10:25
IchKenneDeinenNamenIchKenneDeinenNamen
365
$endgroup$
add a comment |
$begingroup$
I would like to justify the following claim concerning thin sets found in Chung, Lectures from Markov Processes to Brownian Motion, page 112:
Such a set is finely separated in the sense that each of its points has a fine neighborhood containing no other point of the set.
The context is the general theory of Hunt processes. The state space, which is assumed metrizable and compact, is denoted $mathcal{E}_{partial}$.
A (non-necessarily Borel) set $A$ is said to be finely open is for each $x in A$, there is a Borel set $B subset A$ such that $P^{x}[T_{B^c} > 0] = 1$ where $T_{E} equiv inf{ t > 0,;, X_t in E }$ denotes the hitting time of a (nearly-)Borel set $E$ for the Hunt process $(X_t)_{t geq 0}$ and $P^x$ is the probability measure with initial distribution concentrated on $x$. (Hence, if a set $A$ is finely open, then for all $x in A$, $P^x$-almost surely a sample path remains in $A$ for a finite amount of time.) In particular, an open set in $mathcal{E}_{partial}$ is finely open by right-continuity of the sample paths of the Hunt process.
A nearly-Borel set $A$ is said to be thin is for all $x in mathcal{E}_{partial}$, $P^x[T_B > 0] = 1$.
general-topology markov-process stopping-times
asked Dec 18 '18 at 10:25
IchKenneDeinenNamenIchKenneDeinenNamen
365
$endgroup$
I would like to justify the following claim concerning thin sets found in Chung, Lectures from Markov Processes to Brownian Motion, page 112:
Such a set is finely separated in the sense that each of its points has a fine neighborhood containing no other point of the set.
The context is the general theory of Hunt processes. The state space, which is assumed metrizable and compact, is denoted $mathcal{E}_{partial}$.
A (non-necessarily Borel) set $A$ is said to be finely open is for each $x in A$, there is a Borel set $B subset A$ such that $P^{x}[T_{B^c} > 0] = 1$ where $T_{E} equiv inf{ t > 0,;, X_t in E }$ denotes the hitting time of a (nearly-)Borel set $E$ for the Hunt process $(X_t)_{t geq 0}$ and $P^x$ is the probability measure with initial distribution concentrated on $x$. (Hence, if a set $A$ is finely open, then for all $x in A$, $P^x$-almost surely a sample path remains in $A$ for a finite amount of time.) In particular, an open set in $mathcal{E}_{partial}$ is finely open by right-continuity of the sample paths of the Hunt process.
A nearly-Borel set $A$ is said to be thin is for all $x in mathcal{E}_{partial}$, $P^x[T_B > 0] = 1$.
general-topology markov-process stopping-times
general-topology markov-process stopping-times
asked Dec 18 '18 at 10:25
IchKenneDeinenNamenIchKenneDeinenNamen
365
asked Dec 18 '18 at 10:25
IchKenneDeinenNamenIchKenneDeinenNamen
365
asked Dec 18 '18 at 10:25
IchKenneDeinenNamenIchKenneDeinenNamen
365
asked Dec 18 '18 at 10:25
IchKenneDeinenNamenIchKenneDeinenNamen
365
asked Dec 18 '18 at 10:25
IchKenneDeinenNamenIchKenneDeinenNamen
365
365
add a comment |
add a comment |
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