Can $\ell_1(E)$ be embedd into the dual of continuous function space?

Question

Let $E$ be a compact seperable space, for example $E=[0,1]\subset\mathbb{R}$. Denote by $$\ell_1(E):=\{u:=(u_x\in\mathbb{C}:x\in F):F\text{ is a countable subset of }E \text{ and } \|u\|_{\ell_1(E)}:=\sum_{x\in F}|u_x|<\infty\}$$ the Banach space of functions on $E$ that is integrable with respect to the counting measure on $E$. For any $u=(u_1,u_2,\ldots)\in \ell_1(E)$, suppose the corresponding countable subset $F$ is $\{x_1,x_2,\ldots\}\subset E$, then $u$ corresponds to a finite signed Baire measure $$\mu_u(S)=\sum\limits_{x_i\in S\cap F}u_i$$ for any subset $S$ of $E$. With the definition of total variation, we have $$\|\mu_u\|_{TV}=\sum\limits_{i=1}^\infty |u_i|\quad (=\|u\|_{\ell_1(E)})$$ And for any measurable function $f$ on $E$, the integration will be $$\int_E fd\mu_u=\sum\limits_{i=1}^\infty f(x_i)\cdot u_i.$$ By Riesz-Markov-Kakutani Theorem, we know $\mu_u$ is an element of dual space $M(E)$ of continous function space $C(E)$. It seems that we can embed $\ell_1(E)$ into $M(E)$. Is this correct? Is the space $\ell_1(E)$ separable? And do we have some reference of the space $\ell_1(E)$? Thanks a lot!