I'm working on problems relating asymptotic notations. And I stumbled on this easy-looking statement, which asserts that with $S \subseteq \mathbb{Z}$, we have $$\sum_{k \in S} \Theta(f(k)) = \Theta\left(\sum_{k \in S} f(k)\right)$$ There's some necessary clarifications here. On the left-hand side, the notation $\Theta(f(k))$ denotes an anonymous function, say $g(k)$, with the property that $g(k)$ is $\Theta(f(k))$, so it would be clearer to restate the problem as $$\sum_{k \in S} g(k) = \Theta\left(\sum_{k \in S} f(k)\right)$$ At this point, I realized our functions, on the left-hand side and inside the $\Theta$, are not functions that map from $\mathbb{Z}$ (something like $3n^2 + 1 = \Theta(n^2)$, functions just eat the integer $n$), but rather, from the set of subsets of $\mathbb{Z}$, namely the power set $P(\mathbb{Z})$ (like in the assertion, functions eat the whole subset $S$ of $\mathbb{Z}$). But now I don't have any definitions of big-$\Theta$ notation dealing with this kind of functions.
I'm stuck to come up with a way to get through this. Maybe my interpretation of the problem is wrong? I need some help from you guys.