The upper Buck density, introduced by R.C.Buck in 1946, is defined to be the function $$ \mathfrak{b}^\star: \mathcal{P}(\mathbf{N}^+) \to \mathbf{R}: X\mapsto \inf_{X\subseteq A:A\in \mathscr{A}}\mathsf{d}^\star(A), $$ where $\mathscr{A}$ stands for the set of finite unions of arithmetic progressions of $\mathbf{N}^+$ and $\mathsf{d}^\star$ is the classical asymptotic upper density $$ \mathcal{P}(\mathbf{N}^+) \to\mathbf{R}\colon X\mapsto \limsup_{n\to \infty} \frac{|X\cap [1,n]|}{n}. $$ Something on the upper Buck density can be seen, for example, here. Related MO questions concerning the set of upper densities can be found in T1, T2, or T3.

Since $\mathfrak{b}^\star$ is an upper density in this sense, it could be natural to ask whether one can obtain a representation like the one of $\mathsf{d}^\star$; more precisely:

Question. Does there exist a sequence of real-valued functions $(f_n)_{n\ge 1}$ such that each $f_n$ is defined on the power set of $\{1,\ldots,n\}$ and for each $X\subseteq \mathbf{N}^+$ it holds $$ \mathfrak{b}^\star(X)=\limsup_{n\to \infty}f_n(X \cap \{1,\ldots,n\})\,\,\,\,? $$