Completion of a $sigma$-algebra is a $sigma$-algebra












2












$begingroup$



Let $(Omega,mathcal{F},mathbb{P})$ be a probability space. Let
begin{align*}
mathcal{N}&= left{NsubseteqOmega:exists Finmathcal{F}, Nsubseteq F,mathbb{P}(F)=0right} \ mathcal{G}&=left{Acup N:Ainmathcal{F},Ninmathcal{N}right} end{align*}
Prove that $mathcal{G}$ is a $sigma$-algebra.




I've shown that $Omegainmathcal{G}$ and that if $(A_n)_nsubseteqmathcal{G}$ then $cup_n A_ninmathcal{G}$. How can I show that if $Ainmathcal{G}$ then $A^cinmathcal{G}$?



I also noticed $mathcal{F}subseteqmathcal{G}$ if it helps somehow.






probability-theory measure-theory elementary-set-theory







share|cite|improve this question











$endgroup$

















    2












    $begingroup$



    Let $(Omega,mathcal{F},mathbb{P})$ be a probability space. Let
    begin{align*}
    mathcal{N}&= left{NsubseteqOmega:exists Finmathcal{F}, Nsubseteq F,mathbb{P}(F)=0right} \ mathcal{G}&=left{Acup N:Ainmathcal{F},Ninmathcal{N}right} end{align*}
    Prove that $mathcal{G}$ is a $sigma$-algebra.




    I've shown that $Omegainmathcal{G}$ and that if $(A_n)_nsubseteqmathcal{G}$ then $cup_n A_ninmathcal{G}$. How can I show that if $Ainmathcal{G}$ then $A^cinmathcal{G}$?



    I also noticed $mathcal{F}subseteqmathcal{G}$ if it helps somehow.






    probability-theory measure-theory elementary-set-theory







    share|cite|improve this question











    $endgroup$















      2












      2








      2


      1



      $begingroup$



      Let $(Omega,mathcal{F},mathbb{P})$ be a probability space. Let
      begin{align*}
      mathcal{N}&= left{NsubseteqOmega:exists Finmathcal{F}, Nsubseteq F,mathbb{P}(F)=0right} \ mathcal{G}&=left{Acup N:Ainmathcal{F},Ninmathcal{N}right} end{align*}
      Prove that $mathcal{G}$ is a $sigma$-algebra.




      I've shown that $Omegainmathcal{G}$ and that if $(A_n)_nsubseteqmathcal{G}$ then $cup_n A_ninmathcal{G}$. How can I show that if $Ainmathcal{G}$ then $A^cinmathcal{G}$?



      I also noticed $mathcal{F}subseteqmathcal{G}$ if it helps somehow.






      probability-theory measure-theory elementary-set-theory







      share|cite|improve this question











      $endgroup$





      Let $(Omega,mathcal{F},mathbb{P})$ be a probability space. Let
      begin{align*}
      mathcal{N}&= left{NsubseteqOmega:exists Finmathcal{F}, Nsubseteq F,mathbb{P}(F)=0right} \ mathcal{G}&=left{Acup N:Ainmathcal{F},Ninmathcal{N}right} end{align*}
      Prove that $mathcal{G}$ is a $sigma$-algebra.




      I've shown that $Omegainmathcal{G}$ and that if $(A_n)_nsubseteqmathcal{G}$ then $cup_n A_ninmathcal{G}$. How can I show that if $Ainmathcal{G}$ then $A^cinmathcal{G}$?



      I also noticed $mathcal{F}subseteqmathcal{G}$ if it helps somehow.






      probability-theory measure-theory elementary-set-theory




      probability-theory measure-theory elementary-set-theory






      share|cite|improve this question















      share|cite|improve this question













      share|cite|improve this question




      share|cite|improve this question








      edited Jan 22 at 12:35









      Davide Giraudo

      128k17156268




      128k17156268










      asked Jan 18 at 17:42









      J. DoeJ. Doe

      16613




      16613






















          2 Answers
          2






          active

          oldest

          votes


















          3





          +50







          $begingroup$

          Let $G=A cup N$ for $A in mathcal{F}$ and $N in mathcal{N}$. By definition, there exists $F in mathcal{F}$ such that $N subseteq F$ and $mathbb{P}(F)=0$. Clearly,



          $$G^c = A^c cap N^c = underbrace{A^c}_W backslash underbrace{(N cap A^c)}_{U}$$



          As $$underbrace{N cap A^c}_{U} subseteq underbrace{F cap A^c}_{V} subseteq underbrace{A^c}_{W}$$ it follows that



          $$G^c = underbrace{big(A^c backslash (F cap A^c)big)}_{W backslash V} cup underbrace{big( (F cap A^c) backslash (N cap A^c) big)}_{V backslash U}; tag{1}$$



          here we have used the general fact that



          $$U subseteq V subseteq W implies W backslash U = (W backslash V) cup (V backslash U).$$ Consequently, we have shown that



          $$G^c = tilde{A} cup tilde{N} tag{2}$$



          for



          $$tilde{A} := A^c backslash (F cap A^c)quad text{and} quad tilde{N}:= (F cap A^c) backslash (N cap A^c).$$



          As $tilde{N} subseteq F in mathcal{F}$ and $mathbb{P}(F)=0$, we have $tilde{N} in mathcal{N}$. Morover, $A in mathcal{F}$ and $F in mathcal{F}$ implies $tilde{A} in mathcal{F}$. Consequently, $(2)$ shows that $G^c in mathcal{G}$.






          share|cite|improve this answer











          $endgroup$













          • $begingroup$
            Why is $(1)$ true? @saz
            $endgroup$
            – J. Doe
            Jan 22 at 10:26








          • 1




            $begingroup$
            @J.Doe It follows from the "general fact" which I stated directly afterwards.I've just rewritten it a bit... perhaps it's clearer now.
            $endgroup$
            – saz
            Jan 23 at 7:10



















          2












          $begingroup$

          Let $G$ be an event in $mathcal G$, so we can write it in the form $G=Acup N$ with $Ainmathcal F$, $Ninmathcal N$, so $Nsubseteq F$, for some suitable $Fin mathcal F$, $Bbb P(F)=0$.



          Then
          $$
          begin{aligned}
          A &subseteq {color{red}{G}}=Acup Nsubseteq Acup Ftext{ leads to}\
          A^c &supseteq {color{red}{G^c}}=A^ccap N^csupseteq A^ccap F^c ,
          end{aligned}
          $$
          so $G^c$ lies between two events in $mathcal G$ that differ by a null set,
          $$
          0le
          Bbb P(A^c-(A^ccap F^c))
          =
          Bbb P(A^ccap F^{cc})
          =
          Bbb P(A^ccap F)
          le
          Bbb P(F)
          =0
          .
          $$






          share|cite|improve this answer









          $endgroup$













          • $begingroup$
            I got that $A^ccap F^csubseteq G^csubseteq A^c$, but how does $mathbb{P}(A^ccap F)=0$ leads us to $G^cinmathcal{G}$? @dan_fulea
            $endgroup$
            – J. Doe
            Jan 18 at 18:07








          • 1




            $begingroup$
            $G^c$ is then between $A_1:=A^ccap F^c$ and $A_1cup N_1$ for that $N_1$ that makes $A_1cup N_1=A^c$.
            $endgroup$
            – dan_fulea
            Jan 18 at 18:11










          • $begingroup$
            Sorry for stubborning but I still don't get it. We have $A_1=A^ccap F^csubseteq G^csubseteq A^c=A_1cup N_1$, but what does $G^c$ equal to for getting into the set $mathcal{G}$? @dan_fulea
            $endgroup$
            – J. Doe
            Jan 18 at 18:23












          • $begingroup$
            $G^c=A_1cupcolor{red}{(G^c-A_1)}$ and the red entry is a subset of $N_1$. Thanks for asking, please always ask, do not even hesitate to do it. It is then my fault of not finding the argument that hits the point...
            $endgroup$
            – dan_fulea
            Jan 18 at 18:27












          • $begingroup$
            We need that $G^c-A_1=(A^ccap N^c)-(A^ccap F^c)inmathcal{N}$. Why is that true? I observe that $G^c-A_1subseteq A^ccap N^c$ but not sure if it helps. @dan_fulea
            $endgroup$
            – J. Doe
            Jan 19 at 9:59














          Your Answer








          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%2f3078556%2fcompletion-of-a-sigma-algebra-is-a-sigma-algebra%23new-answer', 'question_page');
          }
          );

          Post as a guest















          Required, but never shown

























          2 Answers
          2






          active

          oldest

          votes








          2 Answers
          2






          active

          oldest

          votes









          active

          oldest

          votes






          active

          oldest

          votes









          3





          +50







          $begingroup$

          Let $G=A cup N$ for $A in mathcal{F}$ and $N in mathcal{N}$. By definition, there exists $F in mathcal{F}$ such that $N subseteq F$ and $mathbb{P}(F)=0$. Clearly,



          $$G^c = A^c cap N^c = underbrace{A^c}_W backslash underbrace{(N cap A^c)}_{U}$$



          As $$underbrace{N cap A^c}_{U} subseteq underbrace{F cap A^c}_{V} subseteq underbrace{A^c}_{W}$$ it follows that



          $$G^c = underbrace{big(A^c backslash (F cap A^c)big)}_{W backslash V} cup underbrace{big( (F cap A^c) backslash (N cap A^c) big)}_{V backslash U}; tag{1}$$



          here we have used the general fact that



          $$U subseteq V subseteq W implies W backslash U = (W backslash V) cup (V backslash U).$$ Consequently, we have shown that



          $$G^c = tilde{A} cup tilde{N} tag{2}$$



          for



          $$tilde{A} := A^c backslash (F cap A^c)quad text{and} quad tilde{N}:= (F cap A^c) backslash (N cap A^c).$$



          As $tilde{N} subseteq F in mathcal{F}$ and $mathbb{P}(F)=0$, we have $tilde{N} in mathcal{N}$. Morover, $A in mathcal{F}$ and $F in mathcal{F}$ implies $tilde{A} in mathcal{F}$. Consequently, $(2)$ shows that $G^c in mathcal{G}$.






          share|cite|improve this answer











          $endgroup$













          • $begingroup$
            Why is $(1)$ true? @saz
            $endgroup$
            – J. Doe
            Jan 22 at 10:26








          • 1




            $begingroup$
            @J.Doe It follows from the "general fact" which I stated directly afterwards.I've just rewritten it a bit... perhaps it's clearer now.
            $endgroup$
            – saz
            Jan 23 at 7:10
















          3





          +50







          $begingroup$

          Let $G=A cup N$ for $A in mathcal{F}$ and $N in mathcal{N}$. By definition, there exists $F in mathcal{F}$ such that $N subseteq F$ and $mathbb{P}(F)=0$. Clearly,



          $$G^c = A^c cap N^c = underbrace{A^c}_W backslash underbrace{(N cap A^c)}_{U}$$



          As $$underbrace{N cap A^c}_{U} subseteq underbrace{F cap A^c}_{V} subseteq underbrace{A^c}_{W}$$ it follows that



          $$G^c = underbrace{big(A^c backslash (F cap A^c)big)}_{W backslash V} cup underbrace{big( (F cap A^c) backslash (N cap A^c) big)}_{V backslash U}; tag{1}$$



          here we have used the general fact that



          $$U subseteq V subseteq W implies W backslash U = (W backslash V) cup (V backslash U).$$ Consequently, we have shown that



          $$G^c = tilde{A} cup tilde{N} tag{2}$$



          for



          $$tilde{A} := A^c backslash (F cap A^c)quad text{and} quad tilde{N}:= (F cap A^c) backslash (N cap A^c).$$



          As $tilde{N} subseteq F in mathcal{F}$ and $mathbb{P}(F)=0$, we have $tilde{N} in mathcal{N}$. Morover, $A in mathcal{F}$ and $F in mathcal{F}$ implies $tilde{A} in mathcal{F}$. Consequently, $(2)$ shows that $G^c in mathcal{G}$.






          share|cite|improve this answer











          $endgroup$













          • $begingroup$
            Why is $(1)$ true? @saz
            $endgroup$
            – J. Doe
            Jan 22 at 10:26








          • 1




            $begingroup$
            @J.Doe It follows from the "general fact" which I stated directly afterwards.I've just rewritten it a bit... perhaps it's clearer now.
            $endgroup$
            – saz
            Jan 23 at 7:10














          3





          +50







          3





          +50



          3




          +50



          $begingroup$

          Let $G=A cup N$ for $A in mathcal{F}$ and $N in mathcal{N}$. By definition, there exists $F in mathcal{F}$ such that $N subseteq F$ and $mathbb{P}(F)=0$. Clearly,



          $$G^c = A^c cap N^c = underbrace{A^c}_W backslash underbrace{(N cap A^c)}_{U}$$



          As $$underbrace{N cap A^c}_{U} subseteq underbrace{F cap A^c}_{V} subseteq underbrace{A^c}_{W}$$ it follows that



          $$G^c = underbrace{big(A^c backslash (F cap A^c)big)}_{W backslash V} cup underbrace{big( (F cap A^c) backslash (N cap A^c) big)}_{V backslash U}; tag{1}$$



          here we have used the general fact that



          $$U subseteq V subseteq W implies W backslash U = (W backslash V) cup (V backslash U).$$ Consequently, we have shown that



          $$G^c = tilde{A} cup tilde{N} tag{2}$$



          for



          $$tilde{A} := A^c backslash (F cap A^c)quad text{and} quad tilde{N}:= (F cap A^c) backslash (N cap A^c).$$



          As $tilde{N} subseteq F in mathcal{F}$ and $mathbb{P}(F)=0$, we have $tilde{N} in mathcal{N}$. Morover, $A in mathcal{F}$ and $F in mathcal{F}$ implies $tilde{A} in mathcal{F}$. Consequently, $(2)$ shows that $G^c in mathcal{G}$.






          share|cite|improve this answer











          $endgroup$



          Let $G=A cup N$ for $A in mathcal{F}$ and $N in mathcal{N}$. By definition, there exists $F in mathcal{F}$ such that $N subseteq F$ and $mathbb{P}(F)=0$. Clearly,



          $$G^c = A^c cap N^c = underbrace{A^c}_W backslash underbrace{(N cap A^c)}_{U}$$



          As $$underbrace{N cap A^c}_{U} subseteq underbrace{F cap A^c}_{V} subseteq underbrace{A^c}_{W}$$ it follows that



          $$G^c = underbrace{big(A^c backslash (F cap A^c)big)}_{W backslash V} cup underbrace{big( (F cap A^c) backslash (N cap A^c) big)}_{V backslash U}; tag{1}$$



          here we have used the general fact that



          $$U subseteq V subseteq W implies W backslash U = (W backslash V) cup (V backslash U).$$ Consequently, we have shown that



          $$G^c = tilde{A} cup tilde{N} tag{2}$$



          for



          $$tilde{A} := A^c backslash (F cap A^c)quad text{and} quad tilde{N}:= (F cap A^c) backslash (N cap A^c).$$



          As $tilde{N} subseteq F in mathcal{F}$ and $mathbb{P}(F)=0$, we have $tilde{N} in mathcal{N}$. Morover, $A in mathcal{F}$ and $F in mathcal{F}$ implies $tilde{A} in mathcal{F}$. Consequently, $(2)$ shows that $G^c in mathcal{G}$.







          share|cite|improve this answer














          share|cite|improve this answer



          share|cite|improve this answer








          edited Jan 23 at 7:53

























          answered Jan 21 at 18:51









          sazsaz

          82.7k863132




          82.7k863132












          • $begingroup$
            Why is $(1)$ true? @saz
            $endgroup$
            – J. Doe
            Jan 22 at 10:26








          • 1




            $begingroup$
            @J.Doe It follows from the "general fact" which I stated directly afterwards.I've just rewritten it a bit... perhaps it's clearer now.
            $endgroup$
            – saz
            Jan 23 at 7:10


















          • $begingroup$
            Why is $(1)$ true? @saz
            $endgroup$
            – J. Doe
            Jan 22 at 10:26








          • 1




            $begingroup$
            @J.Doe It follows from the "general fact" which I stated directly afterwards.I've just rewritten it a bit... perhaps it's clearer now.
            $endgroup$
            – saz
            Jan 23 at 7:10
















          $begingroup$
          Why is $(1)$ true? @saz
          $endgroup$
          – J. Doe
          Jan 22 at 10:26






          $begingroup$
          Why is $(1)$ true? @saz
          $endgroup$
          – J. Doe
          Jan 22 at 10:26






          1




          1




          $begingroup$
          @J.Doe It follows from the "general fact" which I stated directly afterwards.I've just rewritten it a bit... perhaps it's clearer now.
          $endgroup$
          – saz
          Jan 23 at 7:10




          $begingroup$
          @J.Doe It follows from the "general fact" which I stated directly afterwards.I've just rewritten it a bit... perhaps it's clearer now.
          $endgroup$
          – saz
          Jan 23 at 7:10











          2












          $begingroup$

          Let $G$ be an event in $mathcal G$, so we can write it in the form $G=Acup N$ with $Ainmathcal F$, $Ninmathcal N$, so $Nsubseteq F$, for some suitable $Fin mathcal F$, $Bbb P(F)=0$.



          Then
          $$
          begin{aligned}
          A &subseteq {color{red}{G}}=Acup Nsubseteq Acup Ftext{ leads to}\
          A^c &supseteq {color{red}{G^c}}=A^ccap N^csupseteq A^ccap F^c ,
          end{aligned}
          $$
          so $G^c$ lies between two events in $mathcal G$ that differ by a null set,
          $$
          0le
          Bbb P(A^c-(A^ccap F^c))
          =
          Bbb P(A^ccap F^{cc})
          =
          Bbb P(A^ccap F)
          le
          Bbb P(F)
          =0
          .
          $$






          share|cite|improve this answer









          $endgroup$













          • $begingroup$
            I got that $A^ccap F^csubseteq G^csubseteq A^c$, but how does $mathbb{P}(A^ccap F)=0$ leads us to $G^cinmathcal{G}$? @dan_fulea
            $endgroup$
            – J. Doe
            Jan 18 at 18:07








          • 1




            $begingroup$
            $G^c$ is then between $A_1:=A^ccap F^c$ and $A_1cup N_1$ for that $N_1$ that makes $A_1cup N_1=A^c$.
            $endgroup$
            – dan_fulea
            Jan 18 at 18:11










          • $begingroup$
            Sorry for stubborning but I still don't get it. We have $A_1=A^ccap F^csubseteq G^csubseteq A^c=A_1cup N_1$, but what does $G^c$ equal to for getting into the set $mathcal{G}$? @dan_fulea
            $endgroup$
            – J. Doe
            Jan 18 at 18:23












          • $begingroup$
            $G^c=A_1cupcolor{red}{(G^c-A_1)}$ and the red entry is a subset of $N_1$. Thanks for asking, please always ask, do not even hesitate to do it. It is then my fault of not finding the argument that hits the point...
            $endgroup$
            – dan_fulea
            Jan 18 at 18:27












          • $begingroup$
            We need that $G^c-A_1=(A^ccap N^c)-(A^ccap F^c)inmathcal{N}$. Why is that true? I observe that $G^c-A_1subseteq A^ccap N^c$ but not sure if it helps. @dan_fulea
            $endgroup$
            – J. Doe
            Jan 19 at 9:59


















          2












          $begingroup$

          Let $G$ be an event in $mathcal G$, so we can write it in the form $G=Acup N$ with $Ainmathcal F$, $Ninmathcal N$, so $Nsubseteq F$, for some suitable $Fin mathcal F$, $Bbb P(F)=0$.



          Then
          $$
          begin{aligned}
          A &subseteq {color{red}{G}}=Acup Nsubseteq Acup Ftext{ leads to}\
          A^c &supseteq {color{red}{G^c}}=A^ccap N^csupseteq A^ccap F^c ,
          end{aligned}
          $$
          so $G^c$ lies between two events in $mathcal G$ that differ by a null set,
          $$
          0le
          Bbb P(A^c-(A^ccap F^c))
          =
          Bbb P(A^ccap F^{cc})
          =
          Bbb P(A^ccap F)
          le
          Bbb P(F)
          =0
          .
          $$






          share|cite|improve this answer









          $endgroup$













          • $begingroup$
            I got that $A^ccap F^csubseteq G^csubseteq A^c$, but how does $mathbb{P}(A^ccap F)=0$ leads us to $G^cinmathcal{G}$? @dan_fulea
            $endgroup$
            – J. Doe
            Jan 18 at 18:07








          • 1




            $begingroup$
            $G^c$ is then between $A_1:=A^ccap F^c$ and $A_1cup N_1$ for that $N_1$ that makes $A_1cup N_1=A^c$.
            $endgroup$
            – dan_fulea
            Jan 18 at 18:11










          • $begingroup$
            Sorry for stubborning but I still don't get it. We have $A_1=A^ccap F^csubseteq G^csubseteq A^c=A_1cup N_1$, but what does $G^c$ equal to for getting into the set $mathcal{G}$? @dan_fulea
            $endgroup$
            – J. Doe
            Jan 18 at 18:23












          • $begingroup$
            $G^c=A_1cupcolor{red}{(G^c-A_1)}$ and the red entry is a subset of $N_1$. Thanks for asking, please always ask, do not even hesitate to do it. It is then my fault of not finding the argument that hits the point...
            $endgroup$
            – dan_fulea
            Jan 18 at 18:27












          • $begingroup$
            We need that $G^c-A_1=(A^ccap N^c)-(A^ccap F^c)inmathcal{N}$. Why is that true? I observe that $G^c-A_1subseteq A^ccap N^c$ but not sure if it helps. @dan_fulea
            $endgroup$
            – J. Doe
            Jan 19 at 9:59
















          2












          2








          2





          $begingroup$

          Let $G$ be an event in $mathcal G$, so we can write it in the form $G=Acup N$ with $Ainmathcal F$, $Ninmathcal N$, so $Nsubseteq F$, for some suitable $Fin mathcal F$, $Bbb P(F)=0$.



          Then
          $$
          begin{aligned}
          A &subseteq {color{red}{G}}=Acup Nsubseteq Acup Ftext{ leads to}\
          A^c &supseteq {color{red}{G^c}}=A^ccap N^csupseteq A^ccap F^c ,
          end{aligned}
          $$
          so $G^c$ lies between two events in $mathcal G$ that differ by a null set,
          $$
          0le
          Bbb P(A^c-(A^ccap F^c))
          =
          Bbb P(A^ccap F^{cc})
          =
          Bbb P(A^ccap F)
          le
          Bbb P(F)
          =0
          .
          $$






          share|cite|improve this answer









          $endgroup$



          Let $G$ be an event in $mathcal G$, so we can write it in the form $G=Acup N$ with $Ainmathcal F$, $Ninmathcal N$, so $Nsubseteq F$, for some suitable $Fin mathcal F$, $Bbb P(F)=0$.



          Then
          $$
          begin{aligned}
          A &subseteq {color{red}{G}}=Acup Nsubseteq Acup Ftext{ leads to}\
          A^c &supseteq {color{red}{G^c}}=A^ccap N^csupseteq A^ccap F^c ,
          end{aligned}
          $$
          so $G^c$ lies between two events in $mathcal G$ that differ by a null set,
          $$
          0le
          Bbb P(A^c-(A^ccap F^c))
          =
          Bbb P(A^ccap F^{cc})
          =
          Bbb P(A^ccap F)
          le
          Bbb P(F)
          =0
          .
          $$







          share|cite|improve this answer












          share|cite|improve this answer



          share|cite|improve this answer










          answered Jan 18 at 17:54









          dan_fuleadan_fulea

          7,1981513




          7,1981513












          • $begingroup$
            I got that $A^ccap F^csubseteq G^csubseteq A^c$, but how does $mathbb{P}(A^ccap F)=0$ leads us to $G^cinmathcal{G}$? @dan_fulea
            $endgroup$
            – J. Doe
            Jan 18 at 18:07








          • 1




            $begingroup$
            $G^c$ is then between $A_1:=A^ccap F^c$ and $A_1cup N_1$ for that $N_1$ that makes $A_1cup N_1=A^c$.
            $endgroup$
            – dan_fulea
            Jan 18 at 18:11










          • $begingroup$
            Sorry for stubborning but I still don't get it. We have $A_1=A^ccap F^csubseteq G^csubseteq A^c=A_1cup N_1$, but what does $G^c$ equal to for getting into the set $mathcal{G}$? @dan_fulea
            $endgroup$
            – J. Doe
            Jan 18 at 18:23












          • $begingroup$
            $G^c=A_1cupcolor{red}{(G^c-A_1)}$ and the red entry is a subset of $N_1$. Thanks for asking, please always ask, do not even hesitate to do it. It is then my fault of not finding the argument that hits the point...
            $endgroup$
            – dan_fulea
            Jan 18 at 18:27












          • $begingroup$
            We need that $G^c-A_1=(A^ccap N^c)-(A^ccap F^c)inmathcal{N}$. Why is that true? I observe that $G^c-A_1subseteq A^ccap N^c$ but not sure if it helps. @dan_fulea
            $endgroup$
            – J. Doe
            Jan 19 at 9:59




















          • $begingroup$
            I got that $A^ccap F^csubseteq G^csubseteq A^c$, but how does $mathbb{P}(A^ccap F)=0$ leads us to $G^cinmathcal{G}$? @dan_fulea
            $endgroup$
            – J. Doe
            Jan 18 at 18:07








          • 1




            $begingroup$
            $G^c$ is then between $A_1:=A^ccap F^c$ and $A_1cup N_1$ for that $N_1$ that makes $A_1cup N_1=A^c$.
            $endgroup$
            – dan_fulea
            Jan 18 at 18:11










          • $begingroup$
            Sorry for stubborning but I still don't get it. We have $A_1=A^ccap F^csubseteq G^csubseteq A^c=A_1cup N_1$, but what does $G^c$ equal to for getting into the set $mathcal{G}$? @dan_fulea
            $endgroup$
            – J. Doe
            Jan 18 at 18:23












          • $begingroup$
            $G^c=A_1cupcolor{red}{(G^c-A_1)}$ and the red entry is a subset of $N_1$. Thanks for asking, please always ask, do not even hesitate to do it. It is then my fault of not finding the argument that hits the point...
            $endgroup$
            – dan_fulea
            Jan 18 at 18:27












          • $begingroup$
            We need that $G^c-A_1=(A^ccap N^c)-(A^ccap F^c)inmathcal{N}$. Why is that true? I observe that $G^c-A_1subseteq A^ccap N^c$ but not sure if it helps. @dan_fulea
            $endgroup$
            – J. Doe
            Jan 19 at 9:59


















          $begingroup$
          I got that $A^ccap F^csubseteq G^csubseteq A^c$, but how does $mathbb{P}(A^ccap F)=0$ leads us to $G^cinmathcal{G}$? @dan_fulea
          $endgroup$
          – J. Doe
          Jan 18 at 18:07






          $begingroup$
          I got that $A^ccap F^csubseteq G^csubseteq A^c$, but how does $mathbb{P}(A^ccap F)=0$ leads us to $G^cinmathcal{G}$? @dan_fulea
          $endgroup$
          – J. Doe
          Jan 18 at 18:07






          1




          1




          $begingroup$
          $G^c$ is then between $A_1:=A^ccap F^c$ and $A_1cup N_1$ for that $N_1$ that makes $A_1cup N_1=A^c$.
          $endgroup$
          – dan_fulea
          Jan 18 at 18:11




          $begingroup$
          $G^c$ is then between $A_1:=A^ccap F^c$ and $A_1cup N_1$ for that $N_1$ that makes $A_1cup N_1=A^c$.
          $endgroup$
          – dan_fulea
          Jan 18 at 18:11












          $begingroup$
          Sorry for stubborning but I still don't get it. We have $A_1=A^ccap F^csubseteq G^csubseteq A^c=A_1cup N_1$, but what does $G^c$ equal to for getting into the set $mathcal{G}$? @dan_fulea
          $endgroup$
          – J. Doe
          Jan 18 at 18:23






          $begingroup$
          Sorry for stubborning but I still don't get it. We have $A_1=A^ccap F^csubseteq G^csubseteq A^c=A_1cup N_1$, but what does $G^c$ equal to for getting into the set $mathcal{G}$? @dan_fulea
          $endgroup$
          – J. Doe
          Jan 18 at 18:23














          $begingroup$
          $G^c=A_1cupcolor{red}{(G^c-A_1)}$ and the red entry is a subset of $N_1$. Thanks for asking, please always ask, do not even hesitate to do it. It is then my fault of not finding the argument that hits the point...
          $endgroup$
          – dan_fulea
          Jan 18 at 18:27






          $begingroup$
          $G^c=A_1cupcolor{red}{(G^c-A_1)}$ and the red entry is a subset of $N_1$. Thanks for asking, please always ask, do not even hesitate to do it. It is then my fault of not finding the argument that hits the point...
          $endgroup$
          – dan_fulea
          Jan 18 at 18:27














          $begingroup$
          We need that $G^c-A_1=(A^ccap N^c)-(A^ccap F^c)inmathcal{N}$. Why is that true? I observe that $G^c-A_1subseteq A^ccap N^c$ but not sure if it helps. @dan_fulea
          $endgroup$
          – J. Doe
          Jan 19 at 9:59






          $begingroup$
          We need that $G^c-A_1=(A^ccap N^c)-(A^ccap F^c)inmathcal{N}$. Why is that true? I observe that $G^c-A_1subseteq A^ccap N^c$ but not sure if it helps. @dan_fulea
          $endgroup$
          – J. Doe
          Jan 19 at 9:59




















          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%2f3078556%2fcompletion-of-a-sigma-algebra-is-a-sigma-algebra%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

          Image
          Human spaceflight From Wikipedia, the free encyclopedia Jump to navigation Jump to search "Space traveler" redirects here. For other uses, see Space traveler (disambiguation). @media all and (max-width:720px){.mw-parser-output .tmulti>.thumbinner{width:100%!important;max-width:none!important}.mw-parser-output .tmulti .tsingle{float:none!important;max-width:none!important;width:100%!important;text-align:center}} Apollo 11 crewmember Buzz Aldrin on the Moon, 1969 International Space Station crewmember Tracy Caldwell Dyson views the Earth, 2010 Space Shuttle Discovery heads into space with a crew aboard, STS-121 in 2006 Inside a space suit on the Canadarm, 1993 Human spaceflight (also referred to as crewed spaceflight or manned spaceflight ) is space travel with a crew or passengers aboard the spacecraft. Spacecraft carrying people may be operated directly, by human crew, or it may be e...
          Read more

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

          Image
          0 1 What exactly does the error message beginning with E: mean? I assume it has to do with the file structure on Ubuntu-on-Windows, but what exactly? The following package was automatically installed and is no longer required: os-prober Use 'apt-get autoremove' to remove it. The following extra packages will be installed: libxslt1.1 The following NEW packages will be installed: libxslt1.1 xmlstarlet 0 upgraded, 2 newly installed, 0 to remove and 0 not upgraded. Need to get 435 kB of archives. After this operation, 1023 kB of additional disk space will be used. Get:1 http://archive.ubuntu.com/ubuntu/ trusty/main libxslt1.1 amd64 1.1.28-2build1 [145 kB] Get:2 http://archive.ubuntu.com/ubuntu/ trusty/universe xmlstarlet amd64 1.5.0-1 [290 kB] ...
          Read more

          張江高科駅

          Image
          張江高科駅 ちょうこうこうか Zhangjiang Hi-Tech Park 所在地 中国上海市浦東新区 所属事業者 上海地鉄運営有限公司（上海軌道交通・駅詳細） 上海浦東現代有軌交通有限公司（駅詳細） テンプレートを表示 張江高科駅 各種表記 繁体字： 張江高科站 簡体字： 张江高科站 拼音 ： Zhàngjiāng Gāokē Zhàn 発音： ヂャンジャン ガオクー ヂャン 英文： Zhangjiang Hi-Tech Park Station テンプレートを表示 張江高科駅 （ちょうこうこうか-えき）は中華人民共和国上海市浦東新区に位置する、上海軌道交通2号線の駅である。 なお、張江高科駅前にある張江軌道電車の 張江地鉄駅駅 についてもここで述べる。 目次 1 利用可能な鉄道路線 2 沿革 3 駅構造 3.1 上海軌道交通 3.2 張江軌道電車 4 駅周辺 5 隣の駅 6 注釈 7 関連項目 利用可能な鉄道路線 上海軌道交通 ■ 2号線 上海浦東現代有軌交通有限公司 ■ 張江軌道電車 沿革 高架駅時代の姿 2000年12月27日 - 2号線の駅が高架駅として開業。 2010年 1月1日 - 張江軌道電車の開業に伴い、張江地鉄駅駅（停留所）を設置。 2月13日 - 駅地下化に伴い高架駅の運用停止。 2月24日 - 10日間の閉鎖を経て地下新駅の開業。 2017年2月9日 - リニューアル工事を経て再開業。乗降客の増加に対応し、ホームへのエレベーター、エスカレーターを増設、券売機エリアを拡張した [1] 。 駅構造 上海軌道交通 上海軌道交通 張江高科駅 5号口 ちょうこうこうか Zhangjiang Hi-Tech Park ◄ 竜陽路 (2.60km) (1.61km) 金科路 ► 所在地 中国上海市浦東新区松涛路 祖冲之路 （中国語版） 北緯31度12分11秒 東経121度34分50秒  /  北緯31.2...
          Read more