Cameron-Martin shifts on infinite Wiener path space ($\frac{d\mathcal{L}(B+h)}{d\mathcal{L}(B)}=\exp \left({\int }_0^{\infty }{g}_{s}d{B}_s-\frac{1}{2}{\int }_0^{\infty }{g}_s^{2}ds\right)$)

If $h(t)={\int }_0^t{g}_{s}ds$ with deterministic $g\in L^2(0,\infty )$, the finite-time Cameron-Martin theorem densities have second moments $e^{{\int }_0^t{g}_s^{2}ds}$ 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 $B_t/t$.