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Suppose $T_1,T_2,\dots$ are the arrival times of a Poisson process with rate $\lambda$ and let $f$ be a non-negative function. Show that $$E\left[ \exp\left(-\sum\limits_{n=1}^\infty f(T_n)\right)\right]=\exp\left(-\lambda\int\limits_0^\infty(1-e^{-f(t)})dt \right)$$ Hint: first show this to be true for simple functions $f$.

I know that you can use any positive function that is the limit of simple functions, but is not as demonstrating this for simple functions

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I am sure you can show the formula if $f=a\mathbf 1_{[s,t]}$ for some $0\leqslant s\lt t$. –  Did Jul 19 '12 at 22:46

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