$X$ be a random variable and $P$ is a probability measure on $\Omega$. Is it true that $$\displaystyle{\int_{\Omega}}X^+\,\mathrm{d}P = \displaystyle{\int _{0}^{\infty}}P(X^+ > t)\,\mathrm{d}t\,, $$
where $X^+(\omega)=\max\{X(\omega),0\}\,$?
I was thinking along the guidelines of Fubini's theorem as probability is a $\sigma$-finite measure. The left hand side of the integral is actually area under a curve taking vertical strips and the right hand side is calculating the same area taking horizontal strips. So by Fubini's theorem they must be equal.
But how do I rigorously show this ?