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Use $\tilde{\mathbb{P}}$ for the reference measure under which $X$ is a Brownian motion. Pathwise, $B_t=X_t-{\int }_0^t{X}_{s}ds$. Put $h_s=X_s{1}_{\{s\le T\}}$ and defineThe stopped integrand has absolute value at most one. The Novikov condition holds on every finite horizon, so $Z$ is a density martingale. The Itô formula also gives the more useful identityThus $Z$ is uniformly integrable over the entire time axis, not just a true martingale on each finite horizon. Under the reference measure, $\tilde{\mathbb{E}}(t\wedge T)=\tilde{\mathbb{E}}{X}_{t\wedge T}^2\le 1$, so $T<\infty$ almost surely. Hence $Z_t\to Z_T>0$ and $\tilde{\mathbb{E}}{Z}_T=1$.

Define the probability measure on the original sigma-algebra byIts restriction to ${\mathcal{F}}_t$ has density $Z_t$. The Girsanov theorem states that subtracting the integrated density integrand from the reference Brownian motion gives a Brownian motion under the new measure. Thereforeis a Brownian motion under $\mathbb{P}$, and $B_{t\wedge T}=W_{t\wedge T}^{\mathbb{P}}$. This is exactly the requested Brownian property until $T$. The bounded Girsanov density for exit of an unstable linear diffusion also proves global absolute continuity, so a mere collection of unspecified finite-horizon measures is unnecessary.