= Solution
The deterministic shift has <derivative> $g_s=-\mathbf1_{[0,1]}(s)$ and finite total energy one. The infinite-horizon Cameron-Martin formula therefore gives
$$
\boxed{\frac{d\mathcal L(B-(\,\cdot\,\wedge1))}{d\mathcal L(B)}(w)=\exp(-w(1)-1/2).}
$$
The integral of this density is one by the moment-generating function of $B_1\sim N(0,1)$. It is strictly positive, so \b[the shifted law and <Wiener measure> are equivalent on the whole half-line]. The density depends only on the path through time one; after that time the shift is constant.
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