It is proven that, for any deterministic $L^2\lbrack 0,1\rbrack$ function $\phi(t)$, $E\bigg(\exp\int^1_0\phi(t)dw_t\bigg\arrowvert \|w\| < \varepsilon\bigg) \rightarrow 1\,\text{as}\,\varepsilon \rightarrow 0,$ where $w_t$ is a standard Brownian motion and $\|\cdot\|$ is any "reasonable" norm on $C_0\lbrack 0,1\rbrack$. Applications to the computation of Onsager-Machlup functionals are pointed out.