After some fruitless efforts to devise a tree-structure based on subset inclusion and after some investigations on the usability of de Bruijn sequences, I finally arrived at a surprisingly simple method, that is also applicable to countably infinite sets of values:

let $\omega(\sigma)\in\mathbb{R};\ \omega(\emptyset):=0$ the function, that assigns a real weight to each subset.

let $(\lbrace a_0\rbrace:=\lbrace\emptyset\rbrace,\lbrace a_1\rbrace,\lbrace a_2\rbrace,\ ...\ )$ be the sequence of sets containing as elements the empty set, resp. the elements of the "base set".

if $(\Omega_{i,0},\ ...\ \Omega_{i,2^{i-1}}) $ denotes the ordered sequence of all subset sums of $(\lbrace a_0\rbrace,\ ...\ \lbrace a_i\rbrace)$

then $(\Omega_{i+1,0},\ ...\ \Omega_{i+1,2^i}) $ is the *merge-sorted union*

$(\Omega_{i,0},\ ...\ \Omega_{i,2^{i-1}}) \cup (\Omega_{i,0}+\omega(\lbrace a_{i+1}\rbrace),\ ...\ \Omega_{i,2^{i-1}}+\omega(\lbrace a_{i+1}\rbrace)) $

The sketched algorithm also indicates, that its complexity is $O(2^n)$
if the base set contains $n$ elements.
If the subset sums are not unique, then duplicates can be skipped without extra effort in the merging process (in that case the index ranges of the sketched algorithm are of course different).

I'm aware that the notation may need improvement; please feel free to edit or suggest improvements.