Let $f : \mathcal{B}\mapsto\mathbb{R}$ be a set function where the domain $\mathcal{B}$ denotes the Borel $\sigma$-algebra, i.e., a collection of all Borel sets.
Is there a source showing $\sup \{ f(B) \ : \ B \in \mathcal{B} \}$ attainable for some $f$? I guess I am looking at a set version of the Weierstrass theorem.
You have to have some condition on $f$. For example, if you denote $|S|$ the cardinality of a set $S$, suppose $f: \mathcal{B} \to \mathbb{R}$ is given by $$f(B) = \begin{cases} 0 & |B| = \infty\\ \frac{|B|}{|B| + 1} & |B| < \infty \end{cases}.$$ Then $\sup_{B \in \mathcal{B}} f(B) = 1$ is not attained.