Let (left(B_{t}ight)_{t geqslant 0}) be a (mathrm{BM}^{1}) and (f in L^{2}left(lambda_{T} otimes mathbb{P}ight.) ) for some (T>0).

Question:

Let $\left(B_{t}ight)_{t \geqslant 0}$ be a $\mathrm{BM}^{1}$ and $f \in L^{2}\left(\lambda_{T} \otimes \mathbb{P}ight.$ ) for some $T>0$. Assume that the limit $\lim _{\epsilon ightarrow 0} f(\epsilon)=f(0)$ exists in $L^{2}(\mathbb{P})$. Show that \[\frac{1}{B_{\epsilon}} \int_{0}^{\epsilon} f(s) d B_{s} \xrightarrow[\epsilon ightarrow 0]{\mathbb{P}} f(0)\]

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