Consider the stochastic process
$$X_t=\exp(-0.5b^2t+bW_t),$$
where $W_t$ is Standard Wiener process defined on a probability space $(\Omega,\mathcal{F}, \mathbb{P})$ and $b\neq0$. I want to evaluate
$$\underset{t\rightarrow \infty}{\lim}\int_{\underline{b}}^{\overline{b}}X_tdb,$$
where $-\infty<\underline{b}<\overline{b}<\infty$. The strong law of large numbers for a Wiener process states $W_t/t \rightarrow 0$ almost surely as $t\rightarrow \infty$ (see e.g. Karatzas-Shreve section 2.9). A direct implication is that $X_t \rightarrow 0$ almost surely. Then if we could interchange the limit and integral, the result would be that the integral approaches 0 almost surely. However, I am not sure if I can do that. Any hints?