= Infinite-horizon singularity of Brownian motion with constant drift
{title2=$P_\mu\perp P_0\ (\mu\ne0)$}
The laws on continuous paths of a standard <Brownian motion> and a <Brownian motion with drift> $\mu\ne0$ are <mutually singular measures> on the whole half-line. The event $\lim_{t\to\infty}\omega(t)/t=0$ has full measure for the former and zero measure for the latter by the <strong law for Brownian motion>. On every fixed finite horizon, their laws are equivalent by the <Girsanov theorem>. The finite-horizon densities $e^{\mu B_t-\mu^2t/2}$ tend to zero <almost surely> while retaining <expectation> one, so they lack <uniform integrability> over all time.
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