Proof that thin sets are finely separated












0












$begingroup$


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 $x in mathcal{E}_{partial}$, $P^x[T_B > 0] = 1$.










share|cite|improve this question









$endgroup$

















    0












    $begingroup$


    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 $x in mathcal{E}_{partial}$, $P^x[T_B > 0] = 1$.










    share|cite|improve this question









    $endgroup$















      0












      0








      0





      $begingroup$


      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 $x in mathcal{E}_{partial}$, $P^x[T_B > 0] = 1$.










      share|cite|improve this question









      $endgroup$




      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






      share|cite|improve this question













      share|cite|improve this question











      share|cite|improve this question




      share|cite|improve this question










      asked Dec 18 '18 at 10:25









      IchKenneDeinenNamenIchKenneDeinenNamen

      295




      295






















          0






          active

          oldest

          votes











          Your Answer





          StackExchange.ifUsing("editor", function () {
          return StackExchange.using("mathjaxEditing", function () {
          StackExchange.MarkdownEditor.creationCallbacks.add(function (editor, postfix) {
          StackExchange.mathjaxEditing.prepareWmdForMathJax(editor, postfix, [["$", "$"], ["\\(","\\)"]]);
          });
          });
          }, "mathjax-editing");

          StackExchange.ready(function() {
          var channelOptions = {
          tags: "".split(" "),
          id: "69"
          };
          initTagRenderer("".split(" "), "".split(" "), channelOptions);

          StackExchange.using("externalEditor", function() {
          // Have to fire editor after snippets, if snippets enabled
          if (StackExchange.settings.snippets.snippetsEnabled) {
          StackExchange.using("snippets", function() {
          createEditor();
          });
          }
          else {
          createEditor();
          }
          });

          function createEditor() {
          StackExchange.prepareEditor({
          heartbeatType: 'answer',
          autoActivateHeartbeat: false,
          convertImagesToLinks: true,
          noModals: true,
          showLowRepImageUploadWarning: true,
          reputationToPostImages: 10,
          bindNavPrevention: true,
          postfix: "",
          imageUploader: {
          brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
          contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
          allowUrls: true
          },
          noCode: true, onDemand: true,
          discardSelector: ".discard-answer"
          ,immediatelyShowMarkdownHelp:true
          });


          }
          });














          draft saved

          draft discarded


















          StackExchange.ready(
          function () {
          StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3044999%2fproof-that-thin-sets-are-finely-separated%23new-answer', 'question_page');
          }
          );

          Post as a guest















          Required, but never shown

























          0






          active

          oldest

          votes








          0






          active

          oldest

          votes









          active

          oldest

          votes






          active

          oldest

          votes
















          draft saved

          draft discarded




















































          Thanks for contributing an answer to Mathematics Stack Exchange!


          • Please be sure to answer the question. Provide details and share your research!

          But avoid



          • Asking for help, clarification, or responding to other answers.

          • Making statements based on opinion; back them up with references or personal experience.


          Use MathJax to format equations. MathJax reference.


          To learn more, see our tips on writing great answers.




          draft saved


          draft discarded














          StackExchange.ready(
          function () {
          StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3044999%2fproof-that-thin-sets-are-finely-separated%23new-answer', 'question_page');
          }
          );

          Post as a guest















          Required, but never shown





















































          Required, but never shown














          Required, but never shown












          Required, but never shown







          Required, but never shown

































          Required, but never shown














          Required, but never shown












          Required, but never shown







          Required, but never shown







          Popular posts from this blog

          Ellipse (mathématiques)

          Quarter-circle Tiles

          Mont Emei