In our analysis course, the following question came up and could, up to now, not be solved:
Let $a: \mathbb{N} \to \mathbb{C}$ be a sequence of complex numbers. What are necessary and sufficient conditions for the existence of a function $\mu: 2^\mathbb{N} \to \mathbb{C}$ satisfying the properties
- $\mu(\{i\}) = a_i$
- $\mu$ is finitely additive, i.e. $A\cap B = \emptyset\implies \mu(A\cup B) = \mu(A)+\mu(B)$?
Partial Results
If we are given such a function defined on a subset of $2^\mathbb{N}$, we can of course extend it to all sets of the form $A\cup B, A\cap B = \emptyset$ and $A\setminus B, B\subset A$. This invites the use of Zorn's Lemma; but it seems impossible to prove that a maximal set closed under these operations must be $2^\mathbb{N}$. However, this approach strongly suggests that $\mu$ exists for all $(a_i)$, as the problem only depends on $2^\mathbb{N}$.
On the other hand, if $a_i$ converges absolutely, one can set $\mu(I) = \sum_{i\in I} a_i$ which fulfills the required properties, but this approach does not generalise at all.