For any $M>0$, let $\mathcal{H}_M(\mathbb{R}^d)$ be the set of all probability density functions with differential entropy less than $M$ in absolute value: $$\mathcal{H}_M(\mathbb{R}^d)=\left\{f\in L^1_+(\mathbb{R}^d);\;\;\left|\int_{\mathbb{R}^d}f\log(f)\,dx\right|<M\right\}.$$ We know that $C_0^\infty$ is dense in $L^1$. Can we say that $C_0^\infty(\mathbb{R}^d)\cap\mathcal{H}_M(\mathbb{R}^d)$ is dense in $\mathcal{H}_M(\mathbb{R}^d)$?
(i.e., given a function $f\in \mathcal{H}_M(\mathbb{R}^d)$, can one always construct a sequence of non-negative functions in $C^\infty_0$ with uniformly bounded differential entropy converging to $f$ ?)
I remember reading in a text about Orlicz spaces that $C^\infty_0$ is dense in the space $$\left\{f\in L^1_+(\mathbb{R}^d);\;\int_{\mathbb{R}^d}f\log(1+f)\,dx<\infty\right\}.$$ Maybe this could be adapted to the previous case?