= Solution
Regard the <space of continuous functions vanishing at infinity> $C_0(\mathbb R^n)$ as a subspace of $L^\infty(\mathbb R^n)$. Point evaluation at the origin,
$$
\ell_0(f)=f(0),
$$
is a bounded linear functional of norm one on this subspace because a continuous function's supremum and essential supremum agree. By the <Hahn-Banach theorem>, $\ell_0$ extends to a bounded linear functional
$$
\ell:L^\infty(\mathbb R^n)\longrightarrow\mathbb C.
$$
Suppose that some $g\in L^1(\mathbb R^n)$ represented this extension:
$$
\ell(f)=\int_{\mathbb R^n}f(x)g(x)\,dx.
$$
Choose $\varphi\in C_c(\mathbb R^n)$ with $0\leq\varphi\leq1$, $\varphi(0)=1$, and support in the unit ball, and set $\varphi_k(x)=\varphi(kx)$. Then $\ell(\varphi_k)=1$ for every $k$. On the other hand, $\varphi_k(x)\to0$ for almost every $x$ and $|\varphi_k g|\leq|g|$. The <dominated convergence theorem> gives
$$
\int\varphi_k(x)g(x)\,dx\longrightarrow0,
$$
a contradiction. Thus $\ell$ is a <singular functional on L infinity> and has no $L^1$ density.
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