function continuous ae but not borel measurable












2














It is easy to prove that if $f$ is a function continuous almost everywhere, then $f$ is Lebesgue-measurable by using the property that $mathcal L$ (the Lebesgue-measure) is complete.



Though I've been wondering if the statement "every function continuous ae is Borel-measurable" is true. I feel like it's not but I have a hard time finding a counterexample. So does such a function (continuous ae and not Borel-measurable) exist?










share|cite|improve this question



























    2














    It is easy to prove that if $f$ is a function continuous almost everywhere, then $f$ is Lebesgue-measurable by using the property that $mathcal L$ (the Lebesgue-measure) is complete.



    Though I've been wondering if the statement "every function continuous ae is Borel-measurable" is true. I feel like it's not but I have a hard time finding a counterexample. So does such a function (continuous ae and not Borel-measurable) exist?










    share|cite|improve this question

























      2












      2








      2







      It is easy to prove that if $f$ is a function continuous almost everywhere, then $f$ is Lebesgue-measurable by using the property that $mathcal L$ (the Lebesgue-measure) is complete.



      Though I've been wondering if the statement "every function continuous ae is Borel-measurable" is true. I feel like it's not but I have a hard time finding a counterexample. So does such a function (continuous ae and not Borel-measurable) exist?










      share|cite|improve this question













      It is easy to prove that if $f$ is a function continuous almost everywhere, then $f$ is Lebesgue-measurable by using the property that $mathcal L$ (the Lebesgue-measure) is complete.



      Though I've been wondering if the statement "every function continuous ae is Borel-measurable" is true. I feel like it's not but I have a hard time finding a counterexample. So does such a function (continuous ae and not Borel-measurable) exist?







      measure-theory lebesgue-measure borel-measures






      share|cite|improve this question













      share|cite|improve this question











      share|cite|improve this question




      share|cite|improve this question










      asked 2 days ago









      Bermudes

      184




      184






















          1 Answer
          1






          active

          oldest

          votes


















          4














          Your assertion is false. Let $E$ be a non-Borel subset of the Cantor set $C$. [There are only $mathfrak c$ Borel subsets of $C$, but $2^{mathfrak c}$ subsets of $C$, so at least one subset is not Borel.] Then the characteristic function $f$ of $E$ is continuous (at least) on the complement of the Cantor set, since $C$ is a closed set. Thus $f$ is continuous a.e. But ${x: f(x)>0} = E$ is not Borel. So $f$ is not Borel measurable.






          share|cite|improve this answer



















          • 1




            Thanks, I tried something like that, but how do you prove that $f$ is continuous on the complement of $C$? I mean if $x in C^complement$, then for any $delta > 0$: $f(x) = inf_{y in (xpmdelta)}f(y) = 0$, but why is it that $lim_{deltato 0}sup_{y in (xpmdelta)}f(y) = 0$?
            – Bermudes
            2 days ago






          • 1




            @Bermudes $f(x)=0$ on the complement of $C$.
            – Yanko
            2 days ago






          • 1




            For every point $a$ not in $C$, there is a neighborhood of $a$ where $f$ is identically zero. This works for any closed set in place of $C$.
            – GEdgar
            2 days ago








          • 1




            Oh sure, $C$ is closed, didn't think of that. Thanks a lot!
            – Bermudes
            2 days ago






          • 1




            I will add that to the answer.
            – GEdgar
            2 days ago











          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%2f3060657%2ffunction-continuous-ae-but-not-borel-measurable%23new-answer', 'question_page');
          }
          );

          Post as a guest















          Required, but never shown

























          1 Answer
          1






          active

          oldest

          votes








          1 Answer
          1






          active

          oldest

          votes









          active

          oldest

          votes






          active

          oldest

          votes









          4














          Your assertion is false. Let $E$ be a non-Borel subset of the Cantor set $C$. [There are only $mathfrak c$ Borel subsets of $C$, but $2^{mathfrak c}$ subsets of $C$, so at least one subset is not Borel.] Then the characteristic function $f$ of $E$ is continuous (at least) on the complement of the Cantor set, since $C$ is a closed set. Thus $f$ is continuous a.e. But ${x: f(x)>0} = E$ is not Borel. So $f$ is not Borel measurable.






          share|cite|improve this answer



















          • 1




            Thanks, I tried something like that, but how do you prove that $f$ is continuous on the complement of $C$? I mean if $x in C^complement$, then for any $delta > 0$: $f(x) = inf_{y in (xpmdelta)}f(y) = 0$, but why is it that $lim_{deltato 0}sup_{y in (xpmdelta)}f(y) = 0$?
            – Bermudes
            2 days ago






          • 1




            @Bermudes $f(x)=0$ on the complement of $C$.
            – Yanko
            2 days ago






          • 1




            For every point $a$ not in $C$, there is a neighborhood of $a$ where $f$ is identically zero. This works for any closed set in place of $C$.
            – GEdgar
            2 days ago








          • 1




            Oh sure, $C$ is closed, didn't think of that. Thanks a lot!
            – Bermudes
            2 days ago






          • 1




            I will add that to the answer.
            – GEdgar
            2 days ago
















          4














          Your assertion is false. Let $E$ be a non-Borel subset of the Cantor set $C$. [There are only $mathfrak c$ Borel subsets of $C$, but $2^{mathfrak c}$ subsets of $C$, so at least one subset is not Borel.] Then the characteristic function $f$ of $E$ is continuous (at least) on the complement of the Cantor set, since $C$ is a closed set. Thus $f$ is continuous a.e. But ${x: f(x)>0} = E$ is not Borel. So $f$ is not Borel measurable.






          share|cite|improve this answer



















          • 1




            Thanks, I tried something like that, but how do you prove that $f$ is continuous on the complement of $C$? I mean if $x in C^complement$, then for any $delta > 0$: $f(x) = inf_{y in (xpmdelta)}f(y) = 0$, but why is it that $lim_{deltato 0}sup_{y in (xpmdelta)}f(y) = 0$?
            – Bermudes
            2 days ago






          • 1




            @Bermudes $f(x)=0$ on the complement of $C$.
            – Yanko
            2 days ago






          • 1




            For every point $a$ not in $C$, there is a neighborhood of $a$ where $f$ is identically zero. This works for any closed set in place of $C$.
            – GEdgar
            2 days ago








          • 1




            Oh sure, $C$ is closed, didn't think of that. Thanks a lot!
            – Bermudes
            2 days ago






          • 1




            I will add that to the answer.
            – GEdgar
            2 days ago














          4












          4








          4






          Your assertion is false. Let $E$ be a non-Borel subset of the Cantor set $C$. [There are only $mathfrak c$ Borel subsets of $C$, but $2^{mathfrak c}$ subsets of $C$, so at least one subset is not Borel.] Then the characteristic function $f$ of $E$ is continuous (at least) on the complement of the Cantor set, since $C$ is a closed set. Thus $f$ is continuous a.e. But ${x: f(x)>0} = E$ is not Borel. So $f$ is not Borel measurable.






          share|cite|improve this answer














          Your assertion is false. Let $E$ be a non-Borel subset of the Cantor set $C$. [There are only $mathfrak c$ Borel subsets of $C$, but $2^{mathfrak c}$ subsets of $C$, so at least one subset is not Borel.] Then the characteristic function $f$ of $E$ is continuous (at least) on the complement of the Cantor set, since $C$ is a closed set. Thus $f$ is continuous a.e. But ${x: f(x)>0} = E$ is not Borel. So $f$ is not Borel measurable.







          share|cite|improve this answer














          share|cite|improve this answer



          share|cite|improve this answer








          edited 2 days ago

























          answered 2 days ago









          GEdgar

          61.9k267168




          61.9k267168








          • 1




            Thanks, I tried something like that, but how do you prove that $f$ is continuous on the complement of $C$? I mean if $x in C^complement$, then for any $delta > 0$: $f(x) = inf_{y in (xpmdelta)}f(y) = 0$, but why is it that $lim_{deltato 0}sup_{y in (xpmdelta)}f(y) = 0$?
            – Bermudes
            2 days ago






          • 1




            @Bermudes $f(x)=0$ on the complement of $C$.
            – Yanko
            2 days ago






          • 1




            For every point $a$ not in $C$, there is a neighborhood of $a$ where $f$ is identically zero. This works for any closed set in place of $C$.
            – GEdgar
            2 days ago








          • 1




            Oh sure, $C$ is closed, didn't think of that. Thanks a lot!
            – Bermudes
            2 days ago






          • 1




            I will add that to the answer.
            – GEdgar
            2 days ago














          • 1




            Thanks, I tried something like that, but how do you prove that $f$ is continuous on the complement of $C$? I mean if $x in C^complement$, then for any $delta > 0$: $f(x) = inf_{y in (xpmdelta)}f(y) = 0$, but why is it that $lim_{deltato 0}sup_{y in (xpmdelta)}f(y) = 0$?
            – Bermudes
            2 days ago






          • 1




            @Bermudes $f(x)=0$ on the complement of $C$.
            – Yanko
            2 days ago






          • 1




            For every point $a$ not in $C$, there is a neighborhood of $a$ where $f$ is identically zero. This works for any closed set in place of $C$.
            – GEdgar
            2 days ago








          • 1




            Oh sure, $C$ is closed, didn't think of that. Thanks a lot!
            – Bermudes
            2 days ago






          • 1




            I will add that to the answer.
            – GEdgar
            2 days ago








          1




          1




          Thanks, I tried something like that, but how do you prove that $f$ is continuous on the complement of $C$? I mean if $x in C^complement$, then for any $delta > 0$: $f(x) = inf_{y in (xpmdelta)}f(y) = 0$, but why is it that $lim_{deltato 0}sup_{y in (xpmdelta)}f(y) = 0$?
          – Bermudes
          2 days ago




          Thanks, I tried something like that, but how do you prove that $f$ is continuous on the complement of $C$? I mean if $x in C^complement$, then for any $delta > 0$: $f(x) = inf_{y in (xpmdelta)}f(y) = 0$, but why is it that $lim_{deltato 0}sup_{y in (xpmdelta)}f(y) = 0$?
          – Bermudes
          2 days ago




          1




          1




          @Bermudes $f(x)=0$ on the complement of $C$.
          – Yanko
          2 days ago




          @Bermudes $f(x)=0$ on the complement of $C$.
          – Yanko
          2 days ago




          1




          1




          For every point $a$ not in $C$, there is a neighborhood of $a$ where $f$ is identically zero. This works for any closed set in place of $C$.
          – GEdgar
          2 days ago






          For every point $a$ not in $C$, there is a neighborhood of $a$ where $f$ is identically zero. This works for any closed set in place of $C$.
          – GEdgar
          2 days ago






          1




          1




          Oh sure, $C$ is closed, didn't think of that. Thanks a lot!
          – Bermudes
          2 days ago




          Oh sure, $C$ is closed, didn't think of that. Thanks a lot!
          – Bermudes
          2 days ago




          1




          1




          I will add that to the answer.
          – GEdgar
          2 days ago




          I will add that to the answer.
          – GEdgar
          2 days ago


















          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.





          Some of your past answers have not been well-received, and you're in danger of being blocked from answering.


          Please pay close attention to the following guidance:


          • 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.


          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%2f3060657%2ffunction-continuous-ae-but-not-borel-measurable%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

          1300-talet

          1300-talet

          Display a custom attribute below product name in the front-end Magento 1.9.3.8