Proof that supremum of an almost surely continuous random function is random variable












1














Let ${X_t, tin[0,T]}$ on {$R, mathfrak B(R) $} be random, almost surely continuous, function. Show that $X^+=sup_{t in[0,T]} X_t$ is random variable.
My proof:

Let $P$ be set points of no continuity of $X_t$.

And let $widetilde X_t =left{
begin{array}{c}
X_t(omega), omegain mathbb Rsetminus P \
0, omegain P
end{array}
right. $



sup$widetilde X_t$ will be finite and: $
{omega: sup_{t in[0,1]}widetilde X_t>x}=bigcup_{tin [0,T]bigcap Q}{omega:widetilde X_t >x}$
Since ${omega:widetilde X_t >x}$ is in $mathfrak B(R)$ then ${omega : sup_{t in[0,1]}widetilde X_t>x}$ in $mathfrak B(R)$.



And since the intervals $(x, +infty)$ form $mathfrak B(R)$,$quad$$widetilde X_t$ is random variable.

Because sup$_{t in[0,T]}widetilde X_t$=sup $_{t in[0,T]}X_t$ $quad$ $forall omegain mathbb Rbackslash P$,$quad$ sup $_{t in[0,T]}X_t$ is random variable too.






probability-theory proof-verification continuity stochastic-processes







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    1














    Let ${X_t, tin[0,T]}$ on {$R, mathfrak B(R) $} be random, almost surely continuous, function. Show that $X^+=sup_{t in[0,T]} X_t$ is random variable.
    My proof:

    Let $P$ be set points of no continuity of $X_t$.

    And let $widetilde X_t =left{
    begin{array}{c}
    X_t(omega), omegain mathbb Rsetminus P \
    0, omegain P
    end{array}
    right. $



    sup$widetilde X_t$ will be finite and: $
    {omega: sup_{t in[0,1]}widetilde X_t>x}=bigcup_{tin [0,T]bigcap Q}{omega:widetilde X_t >x}$
    Since ${omega:widetilde X_t >x}$ is in $mathfrak B(R)$ then ${omega : sup_{t in[0,1]}widetilde X_t>x}$ in $mathfrak B(R)$.



    And since the intervals $(x, +infty)$ form $mathfrak B(R)$,$quad$$widetilde X_t$ is random variable.

    Because sup$_{t in[0,T]}widetilde X_t$=sup $_{t in[0,T]}X_t$ $quad$ $forall omegain mathbb Rbackslash P$,$quad$ sup $_{t in[0,T]}X_t$ is random variable too.






    probability-theory proof-verification continuity stochastic-processes







    share|cite|improve this question



























      1












      1








      1







      Let ${X_t, tin[0,T]}$ on {$R, mathfrak B(R) $} be random, almost surely continuous, function. Show that $X^+=sup_{t in[0,T]} X_t$ is random variable.
      My proof:

      Let $P$ be set points of no continuity of $X_t$.

      And let $widetilde X_t =left{
      begin{array}{c}
      X_t(omega), omegain mathbb Rsetminus P \
      0, omegain P
      end{array}
      right. $



      sup$widetilde X_t$ will be finite and: $
      {omega: sup_{t in[0,1]}widetilde X_t>x}=bigcup_{tin [0,T]bigcap Q}{omega:widetilde X_t >x}$
      Since ${omega:widetilde X_t >x}$ is in $mathfrak B(R)$ then ${omega : sup_{t in[0,1]}widetilde X_t>x}$ in $mathfrak B(R)$.



      And since the intervals $(x, +infty)$ form $mathfrak B(R)$,$quad$$widetilde X_t$ is random variable.

      Because sup$_{t in[0,T]}widetilde X_t$=sup $_{t in[0,T]}X_t$ $quad$ $forall omegain mathbb Rbackslash P$,$quad$ sup $_{t in[0,T]}X_t$ is random variable too.






      probability-theory proof-verification continuity stochastic-processes







      share|cite|improve this question















      Let ${X_t, tin[0,T]}$ on {$R, mathfrak B(R) $} be random, almost surely continuous, function. Show that $X^+=sup_{t in[0,T]} X_t$ is random variable.
      My proof:

      Let $P$ be set points of no continuity of $X_t$.

      And let $widetilde X_t =left{
      begin{array}{c}
      X_t(omega), omegain mathbb Rsetminus P \
      0, omegain P
      end{array}
      right. $



      sup$widetilde X_t$ will be finite and: $
      {omega: sup_{t in[0,1]}widetilde X_t>x}=bigcup_{tin [0,T]bigcap Q}{omega:widetilde X_t >x}$
      Since ${omega:widetilde X_t >x}$ is in $mathfrak B(R)$ then ${omega : sup_{t in[0,1]}widetilde X_t>x}$ in $mathfrak B(R)$.



      And since the intervals $(x, +infty)$ form $mathfrak B(R)$,$quad$$widetilde X_t$ is random variable.

      Because sup$_{t in[0,T]}widetilde X_t$=sup $_{t in[0,T]}X_t$ $quad$ $forall omegain mathbb Rbackslash P$,$quad$ sup $_{t in[0,T]}X_t$ is random variable too.






      probability-theory proof-verification continuity stochastic-processes




      probability-theory proof-verification continuity stochastic-processes






      share|cite|improve this question















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      edited Nov 26 at 22:30









      Davide Giraudo

      125k16150259




      125k16150259










      asked Nov 18 at 15:44









      Emerald

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