The Stone-Cech compactification of $\mathbb{N}$, denoted by $\beta\mathbb{N}$ has the property that every $x\in\ell_\infty$ is identified with (extended uniquely) to $\beta x\in C(\beta\mathbb{N})$.
The dual of $C(\beta\mathbb{N})$ is identified with the space of finite Borel measures on $\beta\mathbb{N}$, that is, by the Riesz's Representation Theorem: – for every $F\in C(\beta\mathbb{N})^*$ there is a unique finite Borel measure $\mu$ on $\beta\mathbb{N}$ such that
$$
\forall f\in C(\beta\mathbb{N}),\ F(f)=\int_{\beta\mathbb{N}} f d\mu
$$
My question is the following conjecture:
Let $\mu\in \ell_\infty^*$. Then $\mu\in\ell_1$ iff $\mu(\beta\mathbb{N}\smallsetminus\mathbb{N})=0$.
Best Answer
For $x\in \ell^\infty,$ let $f_x$ denote the corresponding function on $\beta\mathbb{N}.$ Thus $f_x(n)=x(n)$ for $n\in \mathbb{N}\subset \beta\mathbb{N}.$ For $\mu\in C(\beta \mathbb{N})^*$ let $\varphi_\mu$ denote the corresponding linear functional on $\ell^\infty.$ Assume $\mu$ restricted to $\beta\mathbb{N}\setminus \mathbb{N}$ is zero, i.e. $\mu(A)=0$ for all Borel sets $A \subset \beta\mathbb{N}\setminus \mathbb{N}.$ As $\mu$ is a signed measure with finite variation we get $$\sum_{n=1}^\infty |\mu(n)|<\infty $$ For any $x\in \ell^\infty$ we have $$\varphi_\mu(x)=\mu(f_x)=\int\limits_{\beta \mathbb{N}}f_x(t)\,d\mu(t)=\int\limits_{\mathbb{N}}f_x(t)\,d\mu(t)=\sum_{n=1}^\infty x(n)\mu(n)$$ Hence $\varphi_\mu$ corresponds to $\ell^1$ sequence $\{\mu(n)\}_{n=1}^\infty.$