I have seen this definition of expectation of a nonnegative random variable in many places. I do not quite understand how it is gotten. $$E[X]=\int_0^{\infty}P(X&gt;t) dt $$ I know $E[X]=\int_0^{\infty}x f(x) dx $ This is used to prove the basic version of markov inequality
I'm taking an intro course in Probability theory, and we have just defined expectation for a random variable as $E(X) = E(X^+) - E(X^-)$ if either of them is finite (extending the definition first from simple positive rv to positive rv, and then to all rvs, positive or not).  Then I came across this result for the expectation of a positive random variable. Let $(\Omega, \mathcal{F}, P)$ be a probability space and $X \ge 0$ be a positive rv:  $$ E(X) = \int_{\Omega} X dP = \int_\Omega \int_0^\infty \mathbf{1}_{\{X &gt; t\}} dt  \, d P = \int_{0}^\infty P(X &gt; t) dt. $$ Actually, I'm familiar with this result, and the proof I've seen before involves converting $\int^\infty_0 P(X &gt; t) \, dt$ to a double integral. What I'm struggling with is the step $ \int_{\Omega} X dP = \int_\Omega \int_0^\infty \mathbf{1}_{\{X &gt; t\}} dt  \, d P $, and also the formal definition of an expectation itself.   I have no previous with integration or a "hard analysis" background, and the material covered so far in the course I'm taking has only used $\int_\Omega X dP$ as a notational device for $E(X)$. In general, any pointers to literature or "intuitive" explanations are more than welcome.  Thanks in advance!
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