Completion of $mathbb{E}$ by $mathbb{N}_mu$ regarding Lebesgue measure - are my arguments valid












0












$begingroup$


I have the measure space $(X,mathbb{E},mu)$ where $mathbb{E}$ is a $sigma$-algebra on $X$ and $mu$ is the Lebesgue measure.



I have that
$$mathbb{N}_mu={Esubseteq X; |; text{there exist }Einmathbb{E};text{so};Nsubseteq E;;text{and};mu(E)=0}inmathbb{N}_mu},$$
so is the following argumentation valid:



The enlargement of $mathbb{E}$ with respect to $mu$ is:
$$mathbb{E}_mu=mathbb{E}cupmathbb{N}_mu,$$
so
$$mathbb{E}subseteq mathbb{E}_muquadquadtext{and}quadquadmathbb{N}_musubseteq mathbb{E}_mu$$
as
$mathbb{E}=mathbb{E}_mu$ ${mathbb{N}_mu}$ and $mathbb{N}_mu=mathbb{E}_mu$ ${mathbb{E}}$.






measure-theory lebesgue-measure







share|cite|improve this question











$endgroup$








  • 1




    $begingroup$
    I misunderstood what you did, yes this is basically correct.
    $endgroup$
    – Ian
    Dec 1 '18 at 18:58












  • $begingroup$
    Except there are some places where you are sloppy...for example you really want to add sets $B$ such that there is a measurable set $A$ and a measurable set $C$ with $mu(C)=0$ and $B setminus A subseteq C$.
    $endgroup$
    – Ian
    Dec 1 '18 at 19:11










  • $begingroup$
    your definition of $Bbb N_mu$ seems wrong. I guess that $Bbb N_mu$ want to mean the collection of subsets of any subset of the original $sigma$-algebra of measure zero. If this is the case you must write in the definition $Esubseteq N$ instead of $Nsubseteq E$
    $endgroup$
    – Masacroso
    Dec 1 '18 at 19:28












  • $begingroup$
    there are some more typos, but the assertions are fine besides that. You want to write $Bbb E_musetminusBbb N_mu$ and $Bbb E_musetminusBbb E$ instead of what you wrote
    $endgroup$
    – Masacroso
    Dec 1 '18 at 19:35












  • $begingroup$
    Thank you for your help!
    $endgroup$
    – Frederik
    Dec 2 '18 at 9:30
















0












$begingroup$


I have the measure space $(X,mathbb{E},mu)$ where $mathbb{E}$ is a $sigma$-algebra on $X$ and $mu$ is the Lebesgue measure.



I have that
$$mathbb{N}_mu={Esubseteq X; |; text{there exist }Einmathbb{E};text{so};Nsubseteq E;;text{and};mu(E)=0}inmathbb{N}_mu},$$
so is the following argumentation valid:



The enlargement of $mathbb{E}$ with respect to $mu$ is:
$$mathbb{E}_mu=mathbb{E}cupmathbb{N}_mu,$$
so
$$mathbb{E}subseteq mathbb{E}_muquadquadtext{and}quadquadmathbb{N}_musubseteq mathbb{E}_mu$$
as
$mathbb{E}=mathbb{E}_mu$ ${mathbb{N}_mu}$ and $mathbb{N}_mu=mathbb{E}_mu$ ${mathbb{E}}$.






measure-theory lebesgue-measure







share|cite|improve this question











$endgroup$








  • 1




    $begingroup$
    I misunderstood what you did, yes this is basically correct.
    $endgroup$
    – Ian
    Dec 1 '18 at 18:58












  • $begingroup$
    Except there are some places where you are sloppy...for example you really want to add sets $B$ such that there is a measurable set $A$ and a measurable set $C$ with $mu(C)=0$ and $B setminus A subseteq C$.
    $endgroup$
    – Ian
    Dec 1 '18 at 19:11










  • $begingroup$
    your definition of $Bbb N_mu$ seems wrong. I guess that $Bbb N_mu$ want to mean the collection of subsets of any subset of the original $sigma$-algebra of measure zero. If this is the case you must write in the definition $Esubseteq N$ instead of $Nsubseteq E$
    $endgroup$
    – Masacroso
    Dec 1 '18 at 19:28












  • $begingroup$
    there are some more typos, but the assertions are fine besides that. You want to write $Bbb E_musetminusBbb N_mu$ and $Bbb E_musetminusBbb E$ instead of what you wrote
    $endgroup$
    – Masacroso
    Dec 1 '18 at 19:35












  • $begingroup$
    Thank you for your help!
    $endgroup$
    – Frederik
    Dec 2 '18 at 9:30














0












0








0





$begingroup$


I have the measure space $(X,mathbb{E},mu)$ where $mathbb{E}$ is a $sigma$-algebra on $X$ and $mu$ is the Lebesgue measure.



I have that
$$mathbb{N}_mu={Esubseteq X; |; text{there exist }Einmathbb{E};text{so};Nsubseteq E;;text{and};mu(E)=0}inmathbb{N}_mu},$$
so is the following argumentation valid:



The enlargement of $mathbb{E}$ with respect to $mu$ is:
$$mathbb{E}_mu=mathbb{E}cupmathbb{N}_mu,$$
so
$$mathbb{E}subseteq mathbb{E}_muquadquadtext{and}quadquadmathbb{N}_musubseteq mathbb{E}_mu$$
as
$mathbb{E}=mathbb{E}_mu$ ${mathbb{N}_mu}$ and $mathbb{N}_mu=mathbb{E}_mu$ ${mathbb{E}}$.






measure-theory lebesgue-measure







share|cite|improve this question











$endgroup$




I have the measure space $(X,mathbb{E},mu)$ where $mathbb{E}$ is a $sigma$-algebra on $X$ and $mu$ is the Lebesgue measure.



I have that
$$mathbb{N}_mu={Esubseteq X; |; text{there exist }Einmathbb{E};text{so};Nsubseteq E;;text{and};mu(E)=0}inmathbb{N}_mu},$$
so is the following argumentation valid:



The enlargement of $mathbb{E}$ with respect to $mu$ is:
$$mathbb{E}_mu=mathbb{E}cupmathbb{N}_mu,$$
so
$$mathbb{E}subseteq mathbb{E}_muquadquadtext{and}quadquadmathbb{N}_musubseteq mathbb{E}_mu$$
as
$mathbb{E}=mathbb{E}_mu$ ${mathbb{N}_mu}$ and $mathbb{N}_mu=mathbb{E}_mu$ ${mathbb{E}}$.






measure-theory lebesgue-measure




measure-theory lebesgue-measure






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited Dec 2 '18 at 9:26







Frederik

















asked Dec 1 '18 at 18:40









FrederikFrederik

1009




1009








  • 1




    $begingroup$
    I misunderstood what you did, yes this is basically correct.
    $endgroup$
    – Ian
    Dec 1 '18 at 18:58












  • $begingroup$
    Except there are some places where you are sloppy...for example you really want to add sets $B$ such that there is a measurable set $A$ and a measurable set $C$ with $mu(C)=0$ and $B setminus A subseteq C$.
    $endgroup$
    – Ian
    Dec 1 '18 at 19:11










  • $begingroup$
    your definition of $Bbb N_mu$ seems wrong. I guess that $Bbb N_mu$ want to mean the collection of subsets of any subset of the original $sigma$-algebra of measure zero. If this is the case you must write in the definition $Esubseteq N$ instead of $Nsubseteq E$
    $endgroup$
    – Masacroso
    Dec 1 '18 at 19:28












  • $begingroup$
    there are some more typos, but the assertions are fine besides that. You want to write $Bbb E_musetminusBbb N_mu$ and $Bbb E_musetminusBbb E$ instead of what you wrote
    $endgroup$
    – Masacroso
    Dec 1 '18 at 19:35












  • $begingroup$
    Thank you for your help!
    $endgroup$
    – Frederik
    Dec 2 '18 at 9:30














  • 1




    $begingroup$
    I misunderstood what you did, yes this is basically correct.
    $endgroup$
    – Ian
    Dec 1 '18 at 18:58












  • $begingroup$
    Except there are some places where you are sloppy...for example you really want to add sets $B$ such that there is a measurable set $A$ and a measurable set $C$ with $mu(C)=0$ and $B setminus A subseteq C$.
    $endgroup$
    – Ian
    Dec 1 '18 at 19:11










  • $begingroup$
    your definition of $Bbb N_mu$ seems wrong. I guess that $Bbb N_mu$ want to mean the collection of subsets of any subset of the original $sigma$-algebra of measure zero. If this is the case you must write in the definition $Esubseteq N$ instead of $Nsubseteq E$
    $endgroup$
    – Masacroso
    Dec 1 '18 at 19:28












  • $begingroup$
    there are some more typos, but the assertions are fine besides that. You want to write $Bbb E_musetminusBbb N_mu$ and $Bbb E_musetminusBbb E$ instead of what you wrote
    $endgroup$
    – Masacroso
    Dec 1 '18 at 19:35












  • $begingroup$
    Thank you for your help!
    $endgroup$
    – Frederik
    Dec 2 '18 at 9:30








1




1




$begingroup$
I misunderstood what you did, yes this is basically correct.
$endgroup$
– Ian
Dec 1 '18 at 18:58






$begingroup$
I misunderstood what you did, yes this is basically correct.
$endgroup$
– Ian
Dec 1 '18 at 18:58














$begingroup$
Except there are some places where you are sloppy...for example you really want to add sets $B$ such that there is a measurable set $A$ and a measurable set $C$ with $mu(C)=0$ and $B setminus A subseteq C$.
$endgroup$
– Ian
Dec 1 '18 at 19:11




$begingroup$
Except there are some places where you are sloppy...for example you really want to add sets $B$ such that there is a measurable set $A$ and a measurable set $C$ with $mu(C)=0$ and $B setminus A subseteq C$.
$endgroup$
– Ian
Dec 1 '18 at 19:11












$begingroup$
your definition of $Bbb N_mu$ seems wrong. I guess that $Bbb N_mu$ want to mean the collection of subsets of any subset of the original $sigma$-algebra of measure zero. If this is the case you must write in the definition $Esubseteq N$ instead of $Nsubseteq E$
$endgroup$
– Masacroso
Dec 1 '18 at 19:28






$begingroup$
your definition of $Bbb N_mu$ seems wrong. I guess that $Bbb N_mu$ want to mean the collection of subsets of any subset of the original $sigma$-algebra of measure zero. If this is the case you must write in the definition $Esubseteq N$ instead of $Nsubseteq E$
$endgroup$
– Masacroso
Dec 1 '18 at 19:28














$begingroup$
there are some more typos, but the assertions are fine besides that. You want to write $Bbb E_musetminusBbb N_mu$ and $Bbb E_musetminusBbb E$ instead of what you wrote
$endgroup$
– Masacroso
Dec 1 '18 at 19:35






$begingroup$
there are some more typos, but the assertions are fine besides that. You want to write $Bbb E_musetminusBbb N_mu$ and $Bbb E_musetminusBbb E$ instead of what you wrote
$endgroup$
– Masacroso
Dec 1 '18 at 19:35














$begingroup$
Thank you for your help!
$endgroup$
– Frederik
Dec 2 '18 at 9:30




$begingroup$
Thank you for your help!
$endgroup$
– Frederik
Dec 2 '18 at 9:30










1 Answer
1






active

oldest

votes


















0












$begingroup$

If you start with $(X,mathcal{M},mu)$, the completed $sigma$-algebra is



$$mathcal{M}^*={ B subseteq X : text{ there exists } A,C in mathcal{M} : mu(C)=0 wedge B setminus A subseteq C }.$$



Note that fixing $C=emptyset$ inside that set builder reproduces $mathcal{M}$ itself, so no union is required. Similarly fixing $A=emptyset$ reproduces all of the actual null sets in the completed space.



This is not quite what you wrote, what you wrote has some minor errors that make it rather nonsensical (for example $E$ being bound under two different quantifiers in the same expression).






share|cite|improve this answer











$endgroup$













    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%2f3021667%2fcompletion-of-mathbbe-by-mathbbn-mu-regarding-lebesgue-measure-are%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












    $begingroup$

    If you start with $(X,mathcal{M},mu)$, the completed $sigma$-algebra is



    $$mathcal{M}^*={ B subseteq X : text{ there exists } A,C in mathcal{M} : mu(C)=0 wedge B setminus A subseteq C }.$$



    Note that fixing $C=emptyset$ inside that set builder reproduces $mathcal{M}$ itself, so no union is required. Similarly fixing $A=emptyset$ reproduces all of the actual null sets in the completed space.



    This is not quite what you wrote, what you wrote has some minor errors that make it rather nonsensical (for example $E$ being bound under two different quantifiers in the same expression).






    share|cite|improve this answer











    $endgroup$


















      0












      $begingroup$

      If you start with $(X,mathcal{M},mu)$, the completed $sigma$-algebra is



      $$mathcal{M}^*={ B subseteq X : text{ there exists } A,C in mathcal{M} : mu(C)=0 wedge B setminus A subseteq C }.$$



      Note that fixing $C=emptyset$ inside that set builder reproduces $mathcal{M}$ itself, so no union is required. Similarly fixing $A=emptyset$ reproduces all of the actual null sets in the completed space.



      This is not quite what you wrote, what you wrote has some minor errors that make it rather nonsensical (for example $E$ being bound under two different quantifiers in the same expression).






      share|cite|improve this answer











      $endgroup$
















        0












        0








        0





        $begingroup$

        If you start with $(X,mathcal{M},mu)$, the completed $sigma$-algebra is



        $$mathcal{M}^*={ B subseteq X : text{ there exists } A,C in mathcal{M} : mu(C)=0 wedge B setminus A subseteq C }.$$



        Note that fixing $C=emptyset$ inside that set builder reproduces $mathcal{M}$ itself, so no union is required. Similarly fixing $A=emptyset$ reproduces all of the actual null sets in the completed space.



        This is not quite what you wrote, what you wrote has some minor errors that make it rather nonsensical (for example $E$ being bound under two different quantifiers in the same expression).






        share|cite|improve this answer











        $endgroup$



        If you start with $(X,mathcal{M},mu)$, the completed $sigma$-algebra is



        $$mathcal{M}^*={ B subseteq X : text{ there exists } A,C in mathcal{M} : mu(C)=0 wedge B setminus A subseteq C }.$$



        Note that fixing $C=emptyset$ inside that set builder reproduces $mathcal{M}$ itself, so no union is required. Similarly fixing $A=emptyset$ reproduces all of the actual null sets in the completed space.



        This is not quite what you wrote, what you wrote has some minor errors that make it rather nonsensical (for example $E$ being bound under two different quantifiers in the same expression).







        share|cite|improve this answer














        share|cite|improve this answer



        share|cite|improve this answer








        edited Dec 1 '18 at 19:40

























        answered Dec 1 '18 at 18:46









        IanIan

        67.6k25387




        67.6k25387






























            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%2f3021667%2fcompletion-of-mathbbe-by-mathbbn-mu-regarding-lebesgue-measure-are%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)

            Image
            Pour les articles homonymes, voir Ellipse. L'ellipse est le lieu des points dont la somme des distances à deux points fixes, dits foyers, est constante. Section du cône ou projection du cercle. En géométrie, une ellipse est une courbe plane fermée obtenue par l’intersection d’un cône de révolution avec un plan, à condition que celui-ci coupe l'axe de rotation du cône ou du cylindre : c'est une conique d'excentricité strictement comprise entre 0 et 1. On peut également la définir comme le lieu des points dont la somme des distances à deux points fixes, dits foyers, est constante (sa construction par la méthode du jardinier est très simple). Dans la vie courante, l’ellipse est la forme qu'on perçoit en regardant un cercle en perspective, ou la figure formée par l’ombre d'un disque sur une surface plane. On retrouve aussi, en première approximation [ 1 ] , des ellipses dans les trajectoires des corps célestes (planètes, comètes ou satellites arti
            Read more

            Quarter-circle Tiles

            Image
            up vote 0 down vote favorite I have $99$ identical square tiles, each with a quarter-circle drawn on it like this: [asy] size(1.5cm); draw(Arc((2,0),1,90,180),red+1); draw((0,0)--(2,0)--(2,2)--(0,2)--(0,0)); [/asy] When I arrange the tiles in a $9times 11$ rectangular grid, each with a random orientation, what is the expected value of the number of full circles I form? I think this problem has to do with finding the chance any given 2x2 square has a circle, but I can't find it. expected-value share | cite | improve this question asked Nov 20 at 15:03 6minecraftninja 1 2
            Read more

            Mont Emei

            Image
            Mont Emei.mw-parser-output .entete.map{background-image:url("//upload.wikimedia.org/wikipedia/commons/7/7a/Picto_infobox_map.png")} Vue du sommet Wanfo du mont Emei Géographie Altitude 3 099  m , Wanfo Coordonnées 29° 31′ 11″ nord, 103° 19′ 57″ est Administration Pays   Chine Province Sichuan Ville-préfecture Leshan Géolocalisation sur la carte : Sichuan Mont Emei Géolocalisation sur la carte : Chine Mont Emei modifier   Paysage panoramique du mont Emei, incluant le paysage panoramique du grand Bouddha de Leshan *   Patrimoine mondial de l'UNESCO Passerelle en bois sur le versant ouest Pays   Chine Subdivision Sichuan Type Mixte Critères (iv) (vi) (x) Superficie 15 400 ha Numéro d’identification 779 Zone géographique Asie et Pacifique  ** Année d’inscription 1996 (20 e session) * Descriptif officiel UNESCO ** Classification géog
            Read more