Please help me to show analytically that for A $subseteq$ Ω, the following collection of sets F...












-1












$begingroup$


Please help me to show analytically that for $Asubseteq Ω$, the following collection of sets $F ={emptyset,Omega,A,A^c}$,
is a $sigma$-algebra of subsets of $Omega$.



this is the first time trying this so please help me. I'm really struggling with this matter.



I`m not sure about the tag






real-analysis measure-theory







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closed as unclear what you're asking by Did, Cesareo, mrtaurho, amWhy, José Carlos Santos Jan 4 at 15:46


Please clarify your specific problem or add additional details to highlight exactly what you need. As it's currently written, it’s hard to tell exactly what you're asking. See the How to Ask page for help clarifying this question. If this question can be reworded to fit the rules in the help center, please edit the question.























    -1












    $begingroup$


    Please help me to show analytically that for $Asubseteq Ω$, the following collection of sets $F ={emptyset,Omega,A,A^c}$,
    is a $sigma$-algebra of subsets of $Omega$.



    this is the first time trying this so please help me. I'm really struggling with this matter.



    I`m not sure about the tag






    real-analysis measure-theory







    share|cite|improve this question











    $endgroup$



    closed as unclear what you're asking by Did, Cesareo, mrtaurho, amWhy, José Carlos Santos Jan 4 at 15:46


    Please clarify your specific problem or add additional details to highlight exactly what you need. As it's currently written, it’s hard to tell exactly what you're asking. See the How to Ask page for help clarifying this question. If this question can be reworded to fit the rules in the help center, please edit the question.





















      -1












      -1








      -1





      $begingroup$


      Please help me to show analytically that for $Asubseteq Ω$, the following collection of sets $F ={emptyset,Omega,A,A^c}$,
      is a $sigma$-algebra of subsets of $Omega$.



      this is the first time trying this so please help me. I'm really struggling with this matter.



      I`m not sure about the tag






      real-analysis measure-theory







      share|cite|improve this question











      $endgroup$




      Please help me to show analytically that for $Asubseteq Ω$, the following collection of sets $F ={emptyset,Omega,A,A^c}$,
      is a $sigma$-algebra of subsets of $Omega$.



      this is the first time trying this so please help me. I'm really struggling with this matter.



      I`m not sure about the tag






      real-analysis measure-theory




      real-analysis measure-theory






      share|cite|improve this question















      share|cite|improve this question













      share|cite|improve this question




      share|cite|improve this question








      edited Jan 4 at 7:26









      twnly

      1,0631213




      1,0631213










      asked Jan 4 at 0:39









      Nicolas CloetNicolas Cloet

      11




      11




      closed as unclear what you're asking by Did, Cesareo, mrtaurho, amWhy, José Carlos Santos Jan 4 at 15:46


      Please clarify your specific problem or add additional details to highlight exactly what you need. As it's currently written, it’s hard to tell exactly what you're asking. See the How to Ask page for help clarifying this question. If this question can be reworded to fit the rules in the help center, please edit the question.









      closed as unclear what you're asking by Did, Cesareo, mrtaurho, amWhy, José Carlos Santos Jan 4 at 15:46


      Please clarify your specific problem or add additional details to highlight exactly what you need. As it's currently written, it’s hard to tell exactly what you're asking. See the How to Ask page for help clarifying this question. If this question can be reworded to fit the rules in the help center, please edit the question.
























          1 Answer
          1






          active

          oldest

          votes


















          3












          $begingroup$

          A $sigma$ algebra $F$ of subsets of $Omega$ fulfills the following 4 conditions:



          1) $Omega in F$ (this is fulfilled by your definition of F)



          2) $X in F implies X^c in F$ (which is also trivially fulfilled by your definition)



          3) $(X_n: n in mathbb{N}) in F implies bigcup_{nin mathbb{N}} X_n in F$ (this is also fulfilled as the only subsets aside from $Omega$ and $emptyset$ is the disjoint pair of $A, A^c$).



          4) Finite intersections which is implied by (3).



          And so, $F$ is a $sigma$ algebra of subsets of $Omega$.






          share|cite|improve this answer











          $endgroup$













          • $begingroup$
            Could anyone confirm?
            $endgroup$
            – Nicolas Cloet
            Jan 4 at 1:26






          • 1




            $begingroup$
            You should be able to confirm on your own; nothing here is more complicated than simply stating the definitions.
            $endgroup$
            – Sambo
            Jan 4 at 1:51


















          1 Answer
          1






          active

          oldest

          votes








          1 Answer
          1






          active

          oldest

          votes









          active

          oldest

          votes






          active

          oldest

          votes









          3












          $begingroup$

          A $sigma$ algebra $F$ of subsets of $Omega$ fulfills the following 4 conditions:



          1) $Omega in F$ (this is fulfilled by your definition of F)



          2) $X in F implies X^c in F$ (which is also trivially fulfilled by your definition)



          3) $(X_n: n in mathbb{N}) in F implies bigcup_{nin mathbb{N}} X_n in F$ (this is also fulfilled as the only subsets aside from $Omega$ and $emptyset$ is the disjoint pair of $A, A^c$).



          4) Finite intersections which is implied by (3).



          And so, $F$ is a $sigma$ algebra of subsets of $Omega$.






          share|cite|improve this answer











          $endgroup$













          • $begingroup$
            Could anyone confirm?
            $endgroup$
            – Nicolas Cloet
            Jan 4 at 1:26






          • 1




            $begingroup$
            You should be able to confirm on your own; nothing here is more complicated than simply stating the definitions.
            $endgroup$
            – Sambo
            Jan 4 at 1:51
















          3












          $begingroup$

          A $sigma$ algebra $F$ of subsets of $Omega$ fulfills the following 4 conditions:



          1) $Omega in F$ (this is fulfilled by your definition of F)



          2) $X in F implies X^c in F$ (which is also trivially fulfilled by your definition)



          3) $(X_n: n in mathbb{N}) in F implies bigcup_{nin mathbb{N}} X_n in F$ (this is also fulfilled as the only subsets aside from $Omega$ and $emptyset$ is the disjoint pair of $A, A^c$).



          4) Finite intersections which is implied by (3).



          And so, $F$ is a $sigma$ algebra of subsets of $Omega$.






          share|cite|improve this answer











          $endgroup$













          • $begingroup$
            Could anyone confirm?
            $endgroup$
            – Nicolas Cloet
            Jan 4 at 1:26






          • 1




            $begingroup$
            You should be able to confirm on your own; nothing here is more complicated than simply stating the definitions.
            $endgroup$
            – Sambo
            Jan 4 at 1:51














          3












          3








          3





          $begingroup$

          A $sigma$ algebra $F$ of subsets of $Omega$ fulfills the following 4 conditions:



          1) $Omega in F$ (this is fulfilled by your definition of F)



          2) $X in F implies X^c in F$ (which is also trivially fulfilled by your definition)



          3) $(X_n: n in mathbb{N}) in F implies bigcup_{nin mathbb{N}} X_n in F$ (this is also fulfilled as the only subsets aside from $Omega$ and $emptyset$ is the disjoint pair of $A, A^c$).



          4) Finite intersections which is implied by (3).



          And so, $F$ is a $sigma$ algebra of subsets of $Omega$.






          share|cite|improve this answer











          $endgroup$



          A $sigma$ algebra $F$ of subsets of $Omega$ fulfills the following 4 conditions:



          1) $Omega in F$ (this is fulfilled by your definition of F)



          2) $X in F implies X^c in F$ (which is also trivially fulfilled by your definition)



          3) $(X_n: n in mathbb{N}) in F implies bigcup_{nin mathbb{N}} X_n in F$ (this is also fulfilled as the only subsets aside from $Omega$ and $emptyset$ is the disjoint pair of $A, A^c$).



          4) Finite intersections which is implied by (3).



          And so, $F$ is a $sigma$ algebra of subsets of $Omega$.







          share|cite|improve this answer














          share|cite|improve this answer



          share|cite|improve this answer








          edited Jan 4 at 1:28









          Berci

          61.3k23674




          61.3k23674










          answered Jan 4 at 0:51









          DariusDarius

          1929




          1929












          • $begingroup$
            Could anyone confirm?
            $endgroup$
            – Nicolas Cloet
            Jan 4 at 1:26






          • 1




            $begingroup$
            You should be able to confirm on your own; nothing here is more complicated than simply stating the definitions.
            $endgroup$
            – Sambo
            Jan 4 at 1:51


















          • $begingroup$
            Could anyone confirm?
            $endgroup$
            – Nicolas Cloet
            Jan 4 at 1:26






          • 1




            $begingroup$
            You should be able to confirm on your own; nothing here is more complicated than simply stating the definitions.
            $endgroup$
            – Sambo
            Jan 4 at 1:51
















          $begingroup$
          Could anyone confirm?
          $endgroup$
          – Nicolas Cloet
          Jan 4 at 1:26




          $begingroup$
          Could anyone confirm?
          $endgroup$
          – Nicolas Cloet
          Jan 4 at 1:26




          1




          1




          $begingroup$
          You should be able to confirm on your own; nothing here is more complicated than simply stating the definitions.
          $endgroup$
          – Sambo
          Jan 4 at 1:51




          $begingroup$
          You should be able to confirm on your own; nothing here is more complicated than simply stating the definitions.
          $endgroup$
          – Sambo
          Jan 4 at 1:51



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