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For $X$, a positive random variable, use Fubini's theorem applied to $\sigma$-finite measure to prove

\begin{align*} E(X)= \int_{[0, \infty)} P[X>t] dt \end{align*}

I found several references on the the site like this:

  1. Proving $EX=\int_{0}^{\infty}(1-F(x))\, dx$ when $X\geq0$
  2. Expression for $n$-th moment

but all of them assume that $X$ has density or discrete is the are different proof that does not use this assumption.

This is what I tried

\begin{align*} \int_{[0, \infty)} P[X>t] dt&= \int_{[0, \infty)} E(\mathsf{1}_{[X>t]}) dt=\int_{[0, \infty)} \int_{\Omega} \mathsf{1}_{[X>t]}dP dt\\ &=\int_{\Omega} \int_{[0, \infty)} \mathsf{1}_{[X>t]} dt dP=\int_{\Omega} \int_{[0, X)} 1 dt dP=\int_{\Omega} X dP=E[X] \end{align*} where the first step in second line is due to Fubinni's theorem

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Boby
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    Try to rewrite $P[X>t]$ as an integral of $\chi_{{X>t}}$, then Fubini's theorem will apply naturally. Also observe that $$\int_0^\infty \chi_{{X>t}}(x) dt = \int_0^{X(x)} 1 dt.$$ – Xiao Dec 02 '14 at 23:17
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    Dear, @Xiao I just made editions is this what you meant? – Boby Dec 02 '14 at 23:39
  • Yes, that is what Xiao meant. – copper.hat Dec 02 '14 at 23:46
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    @Boby Nicely done. – Xiao Dec 02 '14 at 23:47
  • @Xiao thanks for you help. If you want please submit an answer and I will give credit it – Boby Dec 02 '14 at 23:48
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    That's okay, glad to help! – Xiao Dec 02 '14 at 23:52
  • Dear @Xiao, I was working on differnt problem http://math.stackexchange.com/questions/1048718/show-that-int-mathbbr-fx-fdx-frac12-if-f-is-a-continuous-dist I think it requires the same approach could you take a look? thanks – Boby Dec 03 '14 at 00:19

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