I know that $C_0^\infty$ is not dense in $BMO$, as the space $C_0$ of continuous functions vanishing at infinity is closed in $VMO\subset BMO$. However, I was wondering if it is dense in $BMO$ with respect to the weak-* topology.
My approach: Let $f\in BMO$ and $g\in\mathcal{H}_a^1$ - a dense subspace of $\mathcal{H}^1$ of bounded functions with compact support. I will approximate $f$ by $f_\varepsilon=f\ast\eta_\varepsilon$, where $\eta_\varepsilon$ is a standard mollifier. By Fubini's theorem we have $$ \int_{\mathbb{R}^n} (f\ast\eta_\varepsilon)g \; \mathrm{d}x = \int_{\mathbb{R}^n} f(g\ast\eta_\varepsilon) \; \mathrm{d}x. $$ Now as $f\in L_{loc}^1$ and $g\in L^\infty$ I can use the fact that $g\ast\eta_\varepsilon$ approximates $g$ in $L^\infty$ with weak-* topology and then $$ \int_{\mathbb{R}^n}f(g\ast\eta_\varepsilon) \; \mathrm{d}x \to \int_{\mathbb{R}^n} fg \; \mathrm{d}x. $$
But here's the key problem: can I conclude from this the convergence for all $g\in\mathcal{H}^1$? As $f_\varepsilon g$ and $fg$ might not necessarily be in $L^1$, the linear functional associated with $f$ is not defined by the integral and I don't know much about it.
Many thanks for any suggestions!