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Suppose that $(X_1, X_2,...)$ is an independent sequence of random variables and $Y$ is measurable $\sigma(X_n,X_{n+1},....)$ for each $n$. Show that there is a constant a such that $P(Y = a) = 1$.

I have tried solving it like this:

The event $(\omega: Y(\omega) \in H)$ $H \in Borel-\sigma$ field is a tail event. By Kolmogorov's 0-1 law all such events have Probability 0 or 1. But I am not able to proceed ahead. Any help?

Artem
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Sriram Natarajan
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  • Is the question maybe a duplicate of this? http://math.stackexchange.com/questions/887782/if-a-probability-space-has-no-measurable-subsets-with-p-strictly-between-0-a – BCLC May 24 '15 at 17:39

1 Answers1

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Consider $$f(t) := P(Y \leq t).$$ It is $\{0,1\}$ valued, increasing, and right continuous. Let $$T := \inf \{t : f(t) = 1\}$$ where $T=\infty$ if the set in question is empty. If $T \in \mathbb{R}$ then by right continuity of $f$ we know $f(T)=1$, and by definition of $T$ we know that $f(t) = 0$ for all $t<T$. It follows that $$P(Y < T) = \lim_{t \nearrow T} P(Y\leq t) = 0.$$ Thus $$P(Y=T) = P(Y\leq T) - P(Y< T) = 1 - 0 = 1.$$

Can you finish up the cases $T=\infty, T=-\infty$?

nullUser
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  • I tried hard but I have my doubts. Why should E[Y] exist? and why should P(Y≥E[Y]),P(Y≤E[Y])>0? Can you give me a detailed explanation? – Sriram Natarajan Apr 29 '15 at 07:23
  • You make a good point, a priori we don't know $E[Y]$ exists. While I was filling in that detail myself I noticed that the proof of the problem is subsumed by the proof that $E[Y]$ exists. (Actually it doesn't need to exist, corresponding to the case $T=\infty, T=-\infty$ above that I have left for you). – nullUser Apr 29 '15 at 15:57
  • Such a simple and powerful idea. I couldn't see this idea for days. Thank you for helping me out – Sriram Natarajan Apr 29 '15 at 16:26
  • Is the question maybe a duplicate of this? http://math.stackexchange.com/questions/887782/if-a-probability-space-has-no-measurable-subsets-with-p-strictly-between-0-a – BCLC May 24 '15 at 17:39