Layer-cake formulas for random variables

For each integer $n\ge 1$, set $X_n=\min \{X,n\}$. This is a finite-valued nonnegative measurable real function: for a real $a$, $\{X_n>a\}$ is $\mathrm{\Omega }$ if $a<0$, is $\{X>a\}$ if $0\le a<n$, and is empty if $a\ge n$. Also $X_n\uparrow X$, including where $X=+\mathrm{\infty }$. Apply [L2] to $X_n$ to obtain $\mathbb{E}[X_n]={\int }_0^{\mathrm{\infty }}\mathbb{P}(X_n>t)dt$. For $t\ge 0$, the tail $\mathbb{P}(X_n>t)$ equals $\mathbb{P}(X>t)$ when $t<n$ and zero when $t\ge n$, so these tails increase pointwise to $\mathbb{P}(X>t)$. Every tail is a nonincreasing Borel function of $t$: each strict upper level set is an initial interval, possibly empty or the whole half-line. Apply [L4] first on $\mathrm{\Omega }$ and then on $(0,\mathrm{\infty })$ to pass to the limits in the identities for $X_n$. Using [L1] gives $\mathbb{E}[X]={\int }_0^{\mathrm{\infty }}\mathbb{P}(X>t)dt,$ with $+\mathrm{\infty }$ allowed.

If $X$ is integrable and real, then $X^{+},X^{-}\in L^1$ and [L1] gives $X=X^{+}-X^{-}$. By step 1.1 applied to $X^{+}$ and $X^{-}$, $\mathbb{E}[X^{+}]={\int }_0^{\mathrm{\infty }}\mathbb{P}(X>t)dt,\mathbb{E}[X^{-}]={\int }_0^{\mathrm{\infty }}\mathbb{P}(X<-t)dt\mathrm{.}$ Subtracting these identities and using [L3] proves the second formula.