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For a positive random variable $X$, show that $ \mathbb{E}(X) = \int_{0}^{\infty} \mathbb{P}(X > t ) \ dt $.

This question has been asked before, and a solution is given here: Integral of CDF equals expected value

However, the proof implicitly uses Fubini's theory for product measures, and I'm looking for a more elementary proof using, say, either the monotone convergence theorem, Fatou's lemma, or dominated convergence. My approach so far has been to try to show that the lim inf and lim sup of "failed" sequences is equal to both sides.

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Peter
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  • Why are you trying to avoid Fubini when it is the natural way to do this? – Potato Jun 22 '13 at 04:11
  • @Potato because the problem is from a chapter before the chapter giving the proof of Fubini's theorem. I'm also interested in it as an application of some other tools, as mentioned. – Peter Jun 22 '13 at 04:13
  • Have you tried proving it for simple functions and using the fact that simple functions are dense in $L^p$? – Potato Jun 22 '13 at 04:15
  • What book, by the way? – Potato Jun 22 '13 at 04:26
  • @Potato Assuming we have it for simple functions, I'm still unsure how to do it. – Peter Jun 22 '13 at 04:38
  • @Potato The book is "The Theory of Probability" by Venkatesh – Peter Jun 22 '13 at 04:38
  • I found a proof without Fubini. It's essentially a "do it for simple functions, then approximate" argument. See pages 197 and 198 of Folland's Real Analysis. – Potato Jun 22 '13 at 04:47
  • http://math.stackexchange.com/a/256006/81456 – bob Jun 22 '13 at 05:26
  • @bob The author there assumes that a density exists and then uses integration by part. I don't necessarily have that a density exists. How would I go about trying to integrate $\int_{0}^{n} X dP $? – Peter Jun 22 '13 at 05:57
  • http://en.wikipedia.org/wiki/Lebesgue%E2%80%93Stieltjes_integration#Integration_by_parts – bob Jun 22 '13 at 06:05

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