Representation of a random variable as a Riemann Integral

Question

Let $X$ be an almost surely non-negative random variable, then is it true that $X=\int_0^\infty 1_{\{X>s\}} ds$? I came across this claim in a proof in Kallenberg's Foundations of Modern Probability. I am guessing that must be trivial exercise but I can't see why it is so so any help will be really appreciated. Will it also be true if $f$ is any measurable numeric function so $f=\int_{-\infty}^\infty 1_{\{X>s\}} ds$?

Thanks in advance.

Answer 1

Since we have $X(\omega)\ge 0$, fixing $\omega$, then we have

$$\int_0^\infty 1_{\{s

For your second question, the answer is no. Consider $f\equiv 1$, you will find the indefinite integral is not well-defined.