 Solution

Let $q$ be the true density and $p$ the announced density. The advantage of truthful reporting for the logarithmic scoring rule isThe Kullback-Leibler divergence is nonnegative, with equality exactly when the densities agree almost everywhere. For instance, Jensen inequality gives ${\mathbb{E}}_q[\log (p/q)]\le \log {\mathbb{E}}_q[p/q]\le 0$; equality requires a constant likelihood ratio on the true support and no remaining mass outside it. ConsequentlyThis proves strict propriety when the expected log scores are well defined, for example with a finite true expected log density. A forecast assigning zero density to a set of positive true probability has score $-\infty$ there and cannot outperform the truth. The hint's strict $>0$ must be corrected to $\ge 0$ to allow its equality case.