Representation of the elements of $c_0^\perp$ as integrals over ultrafilters

Question

Let $$ X=\big\{\varphi\in\ell_\infty^{\,*}(\mathbb N) : \varphi(\{a_n\})=0\,\,\text{whenever $a_n\to 0$}\big\}. $$ If $\varphi_{\mathscr F}(\{a_n\})$ is the limit of $\{a_n\}$ with respect to the non-principal ultrafilter $\mathscr F\in\beta\mathbb N$, then clearly $\varphi_{\mathscr F}\in X$.

I was wondering whether the elements of $X$ can be represented as $$ \varphi=\int_{\beta\mathbb N}\varphi_{\mathscr F}\,d\mu(\mathscr F), $$ where $\mu$ is complex Borel measure on $\beta\mathbb N$ (or perhaps on $\beta\mathbb N\!\setminus\!\mathbb N$).

Is this written somewhere?

Answer 1

This is an expanded version of the previous comment.

In Section 3 (The Stone–Čech compactification of the natural numbers) of the following link (Wikipedia) you can find that $\beta\mathbb{N}$, the Stone–Čech compactification of $\mathbb{N}$, can be seen as the set of all ultrafilters on $\mathbb{N}$, that $\mathbb{N}$ is a dense subset of $\beta\mathbb{N}$ (the trivial ultrafilters), and that $\ell_\infty(\mathbb{N})$ can be identified in a natural way with $C(\beta\mathbb{N})$.

Therefore, by the Riesz representation theorem for $C(K)$ spaces (also called the Riesz–Markov–Kakutani representation theorem), the dual space of $\ell_\infty(\mathbb{N})$ can be identified with $M(\beta\mathbb{N})$, the space of finite Borel measures on $\beta\mathbb{N}$.

Now, for every $F\in \ell_\infty(\mathbb{N})^*$ there exists a measure $\mu_F\in M(\beta\mathbb{N})$ so that, given $x=(x_n)\in \ell_\infty(\mathbb{N})$, we have $F(x)=\int_{\beta\mathbb{N}} f(t) d\mu_F(t)$, where $f\in (\beta\mathbb{N})$ is determined by $f(n)=x_n$ ($n\in\mathbb{N}$).

The functionals $F\in \ell_\infty(\mathbb{N})^*$ which are zero on the subspace $c_0$ ($F\in X$) can be identified with the measures with support contained in $\mathbb{N}^*=\beta\mathbb{N}\setminus\mathbb{N}$, the non-trivial ultrafilters.