Tail integral formula for expectation
= Tail integral formula for expectation
{title2=$\mathbb EY=\int_0^\infty\mathbb P(Y>t)dt$}
For a nonnegative <random variable>, $Y=\int_0^\infty\mathbf1_{\{Y>t\}}dt$ pointwise, including when $Y=\infty$. <Tonelli's theorem> permits taking <expectation> under the integral and proves the formula in the extended nonnegative reals. This relates finiteness of a <Brownian exit time> <expectation> to integrability of its survival probability.