 Integrability of a stopped random-walk increment

Let $X_n$ be integrable independent and identically distributed random variables, and let $T\ge 1$ be a stopping time for their natural filtration, with $\mathbb{E}T<\infty$. Since $\{T\ge n\}$ is measurable before $X_n$ is observed, independence and the Tonelli theorem give $\mathbb{E}|X_T|\le {\sum }_n\mathbb{E}[|X_n|1_{\{T\ge n\}}]=\mathbb{E}|X_1|\mathbb{E}T$. The event $T=n$ need not be independent of $X_n$, so the selected increment can be biased.