Justify that the quantity $\operatorname{Ent}_{\varphi}(g)$ is well defined for all $g \in C^0(\mathbf{R}) \cap CL(\mathbf{R})$ with strictly positive values such that $\int_{-\infty}^{+\infty} g(x)\varphi(x)\,\mathrm{d}x = 1$, where
$$\operatorname{Ent}_{\varphi}(g) = \int_{-\infty}^{+\infty} \ln(g(x))\,g(x)\,\varphi(x)\,\mathrm{d}x.$$
\textit{Hint: You may use question 11.}