Let $L_g$ define the space of all functions such that \begin{align} L_g= \left\{ f: \int g(f(x)) dx <\infty \right\} \end{align} where $g(x)$ is continiously, differentialbe and strictly convex.
Is that the set of bounded and continuous function a dense subset of $L_g$? What other conditions do we need to impose on $g$ to have this.