Is the supremum of an almost surely continuous random function random variable?











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Let {$X_t, tin[0,1]$} on {$R, mathfrak B(R) $} be random, almost surely continuous, function. How to show that $X^+=sup_{t in[0,1]} X_t$ is random variable ?





Perhaps here I can say that $X_t$ it will be a random variable $forall t$ аnd prove the statement like for random variables ?










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    up vote
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    down vote

    favorite












    Let {$X_t, tin[0,1]$} on {$R, mathfrak B(R) $} be random, almost surely continuous, function. How to show that $X^+=sup_{t in[0,1]} X_t$ is random variable ?





    Perhaps here I can say that $X_t$ it will be a random variable $forall t$ аnd prove the statement like for random variables ?










    share|cite|improve this question
























      up vote
      0
      down vote

      favorite









      up vote
      0
      down vote

      favorite











      Let {$X_t, tin[0,1]$} on {$R, mathfrak B(R) $} be random, almost surely continuous, function. How to show that $X^+=sup_{t in[0,1]} X_t$ is random variable ?





      Perhaps here I can say that $X_t$ it will be a random variable $forall t$ аnd prove the statement like for random variables ?










      share|cite|improve this question













      Let {$X_t, tin[0,1]$} on {$R, mathfrak B(R) $} be random, almost surely continuous, function. How to show that $X^+=sup_{t in[0,1]} X_t$ is random variable ?





      Perhaps here I can say that $X_t$ it will be a random variable $forall t$ аnd prove the statement like for random variables ?







      continuity proof-writing stochastic-processes supremum-and-infimum






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      asked Nov 18 at 11:07









      Emerald

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      358






















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          $X^{+}=sup{X_t: 0leq t leq 1,tin mathbb Q}$ outside a null set, so $X^{+}$ is almost everywhere equal to a random variable. So $X^{+}$ is Lebesgue measurable. It need not be measurable w.r.t the Borel sigma field.






          share|cite|improve this answer





















          • Thanks for your answer. But I don't understand, why can we say, that $X^+$ is random variable? I would be grateful for any tips.
            – Emerald
            Nov 18 at 11:55












          • I think you have to look at the source for definitions. A random function is usually defined as a collection of random variables. so it is assumed that each $X_t$ is a random variable. Otherwise there is no hope whatsoever of proving that $X^{+}$ is a random variable.
            – Kavi Rama Murthy
            Nov 18 at 11:59










          • Thank you very much. Now it became clearer.
            – Emerald
            Nov 18 at 12:05











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          1 Answer
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          1 Answer
          1






          active

          oldest

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          active

          oldest

          votes






          active

          oldest

          votes








          up vote
          0
          down vote



          accepted










          $X^{+}=sup{X_t: 0leq t leq 1,tin mathbb Q}$ outside a null set, so $X^{+}$ is almost everywhere equal to a random variable. So $X^{+}$ is Lebesgue measurable. It need not be measurable w.r.t the Borel sigma field.






          share|cite|improve this answer





















          • Thanks for your answer. But I don't understand, why can we say, that $X^+$ is random variable? I would be grateful for any tips.
            – Emerald
            Nov 18 at 11:55












          • I think you have to look at the source for definitions. A random function is usually defined as a collection of random variables. so it is assumed that each $X_t$ is a random variable. Otherwise there is no hope whatsoever of proving that $X^{+}$ is a random variable.
            – Kavi Rama Murthy
            Nov 18 at 11:59










          • Thank you very much. Now it became clearer.
            – Emerald
            Nov 18 at 12:05















          up vote
          0
          down vote



          accepted










          $X^{+}=sup{X_t: 0leq t leq 1,tin mathbb Q}$ outside a null set, so $X^{+}$ is almost everywhere equal to a random variable. So $X^{+}$ is Lebesgue measurable. It need not be measurable w.r.t the Borel sigma field.






          share|cite|improve this answer





















          • Thanks for your answer. But I don't understand, why can we say, that $X^+$ is random variable? I would be grateful for any tips.
            – Emerald
            Nov 18 at 11:55












          • I think you have to look at the source for definitions. A random function is usually defined as a collection of random variables. so it is assumed that each $X_t$ is a random variable. Otherwise there is no hope whatsoever of proving that $X^{+}$ is a random variable.
            – Kavi Rama Murthy
            Nov 18 at 11:59










          • Thank you very much. Now it became clearer.
            – Emerald
            Nov 18 at 12:05













          up vote
          0
          down vote



          accepted







          up vote
          0
          down vote



          accepted






          $X^{+}=sup{X_t: 0leq t leq 1,tin mathbb Q}$ outside a null set, so $X^{+}$ is almost everywhere equal to a random variable. So $X^{+}$ is Lebesgue measurable. It need not be measurable w.r.t the Borel sigma field.






          share|cite|improve this answer












          $X^{+}=sup{X_t: 0leq t leq 1,tin mathbb Q}$ outside a null set, so $X^{+}$ is almost everywhere equal to a random variable. So $X^{+}$ is Lebesgue measurable. It need not be measurable w.r.t the Borel sigma field.







          share|cite|improve this answer












          share|cite|improve this answer



          share|cite|improve this answer










          answered Nov 18 at 11:46









          Kavi Rama Murthy

          43.9k31852




          43.9k31852












          • Thanks for your answer. But I don't understand, why can we say, that $X^+$ is random variable? I would be grateful for any tips.
            – Emerald
            Nov 18 at 11:55












          • I think you have to look at the source for definitions. A random function is usually defined as a collection of random variables. so it is assumed that each $X_t$ is a random variable. Otherwise there is no hope whatsoever of proving that $X^{+}$ is a random variable.
            – Kavi Rama Murthy
            Nov 18 at 11:59










          • Thank you very much. Now it became clearer.
            – Emerald
            Nov 18 at 12:05


















          • Thanks for your answer. But I don't understand, why can we say, that $X^+$ is random variable? I would be grateful for any tips.
            – Emerald
            Nov 18 at 11:55












          • I think you have to look at the source for definitions. A random function is usually defined as a collection of random variables. so it is assumed that each $X_t$ is a random variable. Otherwise there is no hope whatsoever of proving that $X^{+}$ is a random variable.
            – Kavi Rama Murthy
            Nov 18 at 11:59










          • Thank you very much. Now it became clearer.
            – Emerald
            Nov 18 at 12:05
















          Thanks for your answer. But I don't understand, why can we say, that $X^+$ is random variable? I would be grateful for any tips.
          – Emerald
          Nov 18 at 11:55






          Thanks for your answer. But I don't understand, why can we say, that $X^+$ is random variable? I would be grateful for any tips.
          – Emerald
          Nov 18 at 11:55














          I think you have to look at the source for definitions. A random function is usually defined as a collection of random variables. so it is assumed that each $X_t$ is a random variable. Otherwise there is no hope whatsoever of proving that $X^{+}$ is a random variable.
          – Kavi Rama Murthy
          Nov 18 at 11:59




          I think you have to look at the source for definitions. A random function is usually defined as a collection of random variables. so it is assumed that each $X_t$ is a random variable. Otherwise there is no hope whatsoever of proving that $X^{+}$ is a random variable.
          – Kavi Rama Murthy
          Nov 18 at 11:59












          Thank you very much. Now it became clearer.
          – Emerald
          Nov 18 at 12:05




          Thank you very much. Now it became clearer.
          – Emerald
          Nov 18 at 12:05


















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