= Solution
Here is a finite-horizon <Girsanov theorem> statement with the sign fixed explicitly. Let $B$ be a <Brownian motion> under $P$, and let $\theta$ be <predictable> with $\int_0^T\theta_s^2ds<\infty$ almost surely. If
$$
L_t=\exp\!\left(\int_0^t\theta_s dB_s-\frac12\int_0^t\theta_s^2ds\right),\qquad 0\le t\le T,
$$
is a true <martingale> with $\mathbb EL_T=1$, then the measure $Q$ defined by $dQ=L_TdP$ makes $B_t-\int_0^t\theta_sds$ a <Brownian motion> up to $T$. The sufficient <Novikov condition> is $\mathbb E\exp(\tfrac12\int_0^T\theta_s^2ds)<\infty$. Merely being a <local martingale> is not a sufficient density condition.
For a deterministic <absolutely continuous function> $h$ with $h(0)=0$ and <derivative> $g\in L^2[0,T]$, take $\theta=g$. <Novikov's condition> holds because its exponential is deterministic and finite. Under $Q$, the canonical process has the law of a <Brownian motion> plus $h$. Thus for every bounded measurable path functional $F$,
$$
\boxed{\mathbb E F(B+h)=\mathbb E\left[F(B)\exp\!\left(\int_0^Tg_s dB_s-\frac12\int_0^Tg_s^2ds\right)\right].}
$$
This derives the <Cameron-Martin theorem> density formula from the <Girsanov theorem>, rather than assuming the two measures have the same drift convention. Replacing $h$ by $-h$ changes the sign of the <stochastic integral>.
The question's path space is the whole half-line, not just one finite horizon. When $g\in L^2(0,\infty)$, the density <martingale> satisfies
$$
\mathbb EL_t^2=\exp\!\left(\int_0^tg_s^2ds\right)\le e^{\|g\|_2^2}.
$$
It has a <uniformly integrable> limit $L_\infty=\exp(\int_0^\infty g_s dB_s-\|g\|_2^2/2)$ of mean one, strictly positive almost surely. Integrals here converge in $L^2$ and almost surely. The finite-time identities then identify $L_\infty$ as the density on the full path <sigma-algebra>, since finite-time cylinder events generate it. This is <Cameron-Martin shifts on infinite Wiener path space>. Finite-horizon <absolute continuity of measures> alone does not imply infinite-horizon <absolute continuity of measures>.
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