= Solution
The expression obtained in part (a)(iii) gives a direct deterministic bound. For every fixed $T<\infty$ and $0\leq s\leq T$,
$$
0<Z_s
=\exp\!\left[\frac{\beta s}{2}-\frac\beta2B_s^2
-\frac{\beta^2}{2}\int_0^sB_u^2du\right]
\leq e^{\beta T/2}.
$$
Thus the <bounded local martingale criterion> makes $Z$ a true <martingale> on every finite time interval. More explicitly, if $\sigma_n$ is a <localizing sequence>, the identities $\mathbb E[Z_{t\wedge\sigma_n}\mid\mathcal F_s]=Z_{s\wedge\sigma_n}$ pass to the limit by <dominated convergence>, using the displayed bound for $t\leq T$. Therefore
$$
\boxed{\mathbb E[Z_t\mid\mathcal F_s]=Z_s,\qquad \mathbb EZ_t=1\quad(0\leq s\leq t<\infty).}
$$
This establishes exactly the finite-horizon property needed for the subsequent change of measure.
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