Let $J \subseteq \mathcal{X}$ be an arbitrary classification region. Write the associated risk as \begin{align*} R_\pi(\delta_J) &= \pi_1 \int_{J^c} f_1(x)\, dx + (1 - \pi_1) \int_J f_0(x)\, dx \\ &= \int_{J^c} [\pi_1 f_1(x) - (1-\pi_1) f_0(x)]\, dx + (1-\pi_1)\int_{\mathcal{X}} f_0(x)\, dx. \end{align*} The second term is independent of $J$. The first term is minimized by integrating over the region $J^c$ where the integrand $\pi_1 f_1(x) - (1-\pi_1) f_0(x)$ is as negative as possible, which is achieved by placing in $J$ exactly the points where $\pi_1 f_1(x) - (1-\pi_1) f_0(x) \geq 0$, i.e., where the likelihood ratio is at least $1$. This is precisely the Bayes critical region $R$. Uniqueness holds when the boundary $\{x : \pi_1 f_1(x) = (1-\pi_1) f_0(x)\}$ has probability zero under both $P_0$ and $P_1$, since then reassigning boundary points does not affect the value of the integral.