This is Exercise 14.30 from Probability for Statistics and Machine Learning.
Let $X_i$ be iid with $E|X| < \infty$, and let $T$ be a stopping time adapted to $\{ X_i \}$. Let $S_n = \sum_{i=1}^{n} X_i$. If $E|S_T| = \infty$, then $E[T] = \infty$ as well.
I'm not really sure where to start. Clearly, $E[T] = \sum_{t \geq 1} t P(T = t)$, but how do I relate $P(T=t)$ to $S_T$ when I know nothing about the stopping time explicitly?