= Solution
The <Exponential martingale for Brownian motion> is a true <martingale>: independent increments and the <normal distribution> identity $\mathbb E e^{aN}=e^{a^2v/2}$ for $N\sim N(0,v)$ give $\mathbb E[Z_t\mid\mathcal F_s]=Z_s$. Its second moment is
$$
\mathbb E Z_t^2=e^{\mu^2t},\qquad
\boxed{\sup_{0\leq t\leq T}\mathbb E Z_t^2=e^{\mu^2T}<\infty.}
$$
The <uniform integrability from bounded second moments> criterion now applies: for $K>0$,
$$
\sup_{t\leq T}\mathbb E[Z_t\mathbf1_{\{Z_t>K\}}]\leq\frac{e^{\mu^2T}}K\longrightarrow0.
$$
Thus $\{Z_t:0\leq t\leq T\}$ has <uniform integrability>. In fact, all values stopped at <stopping times> bounded by $T$ have <uniform integrability>, since $Z_\tau=\mathbb E[Z_T\mid\mathcal F_\tau]$ and <uniform integrability of conditional expectations> applies. This is a finite-horizon assertion, not an assertion of <uniform integrability> over all $t\geq0$.
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