Simple question about convergence a.e












2














How could following property be proved without using Dominated Convergence Theorem?



Let $left{{f_n(x)}right}$ ($f_n:mathbb{R^m}rightarrow{}mathbb{R}$) be a sequence of functions such that $f_nrightarrow{f};a.e$ and $|f_n|leq{g}$ with $g$ an integrable function.
Then $f_nrightarrow{f}$ in $L_1(mathbb{R^n})$.



Does someone know a counterexample if it isn't true the second condition?



Thanks.










share|cite|improve this question
























  • Your question basically is: Prove the dominated convergence theorem without using the dominated convergence theorem.
    – Math_QED
    Dec 29 '18 at 10:57










  • Ok ok. I understand
    – mathlife
    Dec 29 '18 at 11:02










  • So for a proof, just look at any basic measure theory book (Rudin's POMA/RCA - Folland's Real analysis - Royden's Real analysis are examples that come to mind).
    – Math_QED
    Dec 29 '18 at 11:05
















2














How could following property be proved without using Dominated Convergence Theorem?



Let $left{{f_n(x)}right}$ ($f_n:mathbb{R^m}rightarrow{}mathbb{R}$) be a sequence of functions such that $f_nrightarrow{f};a.e$ and $|f_n|leq{g}$ with $g$ an integrable function.
Then $f_nrightarrow{f}$ in $L_1(mathbb{R^n})$.



Does someone know a counterexample if it isn't true the second condition?



Thanks.










share|cite|improve this question
























  • Your question basically is: Prove the dominated convergence theorem without using the dominated convergence theorem.
    – Math_QED
    Dec 29 '18 at 10:57










  • Ok ok. I understand
    – mathlife
    Dec 29 '18 at 11:02










  • So for a proof, just look at any basic measure theory book (Rudin's POMA/RCA - Folland's Real analysis - Royden's Real analysis are examples that come to mind).
    – Math_QED
    Dec 29 '18 at 11:05














2












2








2







How could following property be proved without using Dominated Convergence Theorem?



Let $left{{f_n(x)}right}$ ($f_n:mathbb{R^m}rightarrow{}mathbb{R}$) be a sequence of functions such that $f_nrightarrow{f};a.e$ and $|f_n|leq{g}$ with $g$ an integrable function.
Then $f_nrightarrow{f}$ in $L_1(mathbb{R^n})$.



Does someone know a counterexample if it isn't true the second condition?



Thanks.










share|cite|improve this question















How could following property be proved without using Dominated Convergence Theorem?



Let $left{{f_n(x)}right}$ ($f_n:mathbb{R^m}rightarrow{}mathbb{R}$) be a sequence of functions such that $f_nrightarrow{f};a.e$ and $|f_n|leq{g}$ with $g$ an integrable function.
Then $f_nrightarrow{f}$ in $L_1(mathbb{R^n})$.



Does someone know a counterexample if it isn't true the second condition?



Thanks.







real-analysis functional-analysis analysis






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited Dec 29 '18 at 10:43









mathcounterexamples.net

25.3k21953




25.3k21953










asked Dec 29 '18 at 10:32









mathlifemathlife

629




629












  • Your question basically is: Prove the dominated convergence theorem without using the dominated convergence theorem.
    – Math_QED
    Dec 29 '18 at 10:57










  • Ok ok. I understand
    – mathlife
    Dec 29 '18 at 11:02










  • So for a proof, just look at any basic measure theory book (Rudin's POMA/RCA - Folland's Real analysis - Royden's Real analysis are examples that come to mind).
    – Math_QED
    Dec 29 '18 at 11:05


















  • Your question basically is: Prove the dominated convergence theorem without using the dominated convergence theorem.
    – Math_QED
    Dec 29 '18 at 10:57










  • Ok ok. I understand
    – mathlife
    Dec 29 '18 at 11:02










  • So for a proof, just look at any basic measure theory book (Rudin's POMA/RCA - Folland's Real analysis - Royden's Real analysis are examples that come to mind).
    – Math_QED
    Dec 29 '18 at 11:05
















Your question basically is: Prove the dominated convergence theorem without using the dominated convergence theorem.
– Math_QED
Dec 29 '18 at 10:57




Your question basically is: Prove the dominated convergence theorem without using the dominated convergence theorem.
– Math_QED
Dec 29 '18 at 10:57












Ok ok. I understand
– mathlife
Dec 29 '18 at 11:02




Ok ok. I understand
– mathlife
Dec 29 '18 at 11:02












So for a proof, just look at any basic measure theory book (Rudin's POMA/RCA - Folland's Real analysis - Royden's Real analysis are examples that come to mind).
– Math_QED
Dec 29 '18 at 11:05




So for a proof, just look at any basic measure theory book (Rudin's POMA/RCA - Folland's Real analysis - Royden's Real analysis are examples that come to mind).
– Math_QED
Dec 29 '18 at 11:05










1 Answer
1






active

oldest

votes


















0














The property you are stating is the Dominated Convergence theorem: https://en.wikipedia.org/wiki/Dominated_convergence_theorem



A simple counter example is the sequence of functions $f_n: mathbb R to mathbb R$ defined by $$f_n(x) := mathbb 1_{[n,n+1[}.$$
Then $f_n to 0$ pointwise, but $$intlvert f_n - 0 rvert , dlambda = 1 quad forall nin mathbb N.$$
Note that there does not exist an integrable $g$ as in the statement of your property.






share|cite|improve this answer























    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%2f3055715%2fsimple-question-about-convergence-a-e%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









    0














    The property you are stating is the Dominated Convergence theorem: https://en.wikipedia.org/wiki/Dominated_convergence_theorem



    A simple counter example is the sequence of functions $f_n: mathbb R to mathbb R$ defined by $$f_n(x) := mathbb 1_{[n,n+1[}.$$
    Then $f_n to 0$ pointwise, but $$intlvert f_n - 0 rvert , dlambda = 1 quad forall nin mathbb N.$$
    Note that there does not exist an integrable $g$ as in the statement of your property.






    share|cite|improve this answer




























      0














      The property you are stating is the Dominated Convergence theorem: https://en.wikipedia.org/wiki/Dominated_convergence_theorem



      A simple counter example is the sequence of functions $f_n: mathbb R to mathbb R$ defined by $$f_n(x) := mathbb 1_{[n,n+1[}.$$
      Then $f_n to 0$ pointwise, but $$intlvert f_n - 0 rvert , dlambda = 1 quad forall nin mathbb N.$$
      Note that there does not exist an integrable $g$ as in the statement of your property.






      share|cite|improve this answer


























        0












        0








        0






        The property you are stating is the Dominated Convergence theorem: https://en.wikipedia.org/wiki/Dominated_convergence_theorem



        A simple counter example is the sequence of functions $f_n: mathbb R to mathbb R$ defined by $$f_n(x) := mathbb 1_{[n,n+1[}.$$
        Then $f_n to 0$ pointwise, but $$intlvert f_n - 0 rvert , dlambda = 1 quad forall nin mathbb N.$$
        Note that there does not exist an integrable $g$ as in the statement of your property.






        share|cite|improve this answer














        The property you are stating is the Dominated Convergence theorem: https://en.wikipedia.org/wiki/Dominated_convergence_theorem



        A simple counter example is the sequence of functions $f_n: mathbb R to mathbb R$ defined by $$f_n(x) := mathbb 1_{[n,n+1[}.$$
        Then $f_n to 0$ pointwise, but $$intlvert f_n - 0 rvert , dlambda = 1 quad forall nin mathbb N.$$
        Note that there does not exist an integrable $g$ as in the statement of your property.







        share|cite|improve this answer














        share|cite|improve this answer



        share|cite|improve this answer








        edited Dec 29 '18 at 10:52

























        answered Dec 29 '18 at 10:46









        bavor42bavor42

        30419




        30419






























            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%2f3055715%2fsimple-question-about-convergence-a-e%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

            Human spaceflight

            Can not write log (Is /dev/pts mounted?) - openpty in Ubuntu-on-Windows?

            張江高科駅