= Cameron-Martin shifts on infinite Wiener path space
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{title2=$\frac{d\mathcal L(B+h)}{d\mathcal L(B)}=\exp\!\left(\int_0^\infty g_s\,dB_s-\frac12\int_0^\infty g_s^2ds\right)$}
If $h(t)=\int_0^tg_sds$ with deterministic $g\in L^2(0,\infty)$, the finite-time <Cameron-Martin theorem> densities have second moments $e^{\int_0^tg_s^2ds}$ bounded uniformly in time. Their <uniformly integrable martingale> limit is the strictly positive displayed density, giving equivalence on the full infinite continuous-path space. Finite-time densities alone do not guarantee such a global density: a nonzero constant drift has infinite energy and its infinite-horizon path law is singular, distinguished by the limiting ratio $B_t/t$.
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