Showing that $H \geq 0$ implies $E[H] \geq 0$

Question

I am considering the following formula $E[H(\tau)] = pH(0) + \int_{0}^{\infty} H(t)f(t) dt,$ where $0 \leq p \leq 1$, $\int_{0}^{\infty} f(t) dt = 1-p$ and $H$ is a function over which this formula is defined. I wish to show that

if $H \geq 0$, then $E[H(\tau)] \geq 0$,

but I know not how. Evidently, $pH(0) \geq 0$, but how can I argue that the second term is positive?