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
$$
\mathcal{P}(\mathbf{N}^+) \to\mathbf{R}\colon X\mapsto \limsup_{n\to \infty} \frac{|X\cap [1,n]|}{n}.
$$
The upper Buck density has been studied in literature, see e.g. [here][1] or [here][2]. Related MO questions concerning the set of upper densities can be found in [T1][3], [T2][4], or [T3][5].

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
$$
\mathfrak{b}^\star(X)=\limsup_{n\to \infty}f_n(X \cap \{1,\ldots,n\})\,\,\,\,?
$$



  [1]: http://dml.cz/bitstream/handle/10338.dmlcz/136532/MathSlov_41-1991-3_8.pdf
  [2]: http://dml.cz/bitstream/handle/10338.dmlcz/128687/MathSlov_42-1992-1_2.pdf
  [3]: http://mathoverflow.net/questions/214878/additivity-of-upper-densities-with-respect-to-arithmetic-progressions-of-integer
  [4]: http://mathoverflow.net/questions/226512/are-the-extremal-points-of-a-certain-set-of-functions-mathcal-p-mathbf-n-to?lq=1
  [5]: http://mathoverflow.net/questions/227293/superadditivity-of-the-lower-density