Summing Over Uncountable Index Sets

Question

In answering the question Why do we classify infinities in so many symbols and ideas?, William's answer asserted that summing over an uncountable index set necessarily results in an infinite sum. I am curious about whether this is true and, if so, if it is possible to characterise the transfinite nature of the resultant sums over uncountable index sets.

Is it possible to conjecture, e.g., that sums over index sets of $\aleph_{n}$ result in sums of cardinality $\aleph_{n}$? At most $\aleph_{n}$? Between $\aleph_{n-1}$ and $\aleph_{n}$?

Answer 1

Summing over an uncountable index results in an infinite sum if uncountably many terms are non-zero.

To see this, we prove the contrapositive: that if a sum over an uncountable index is finite implies that at most countably many terms are non-zero.

Proof. Let $\sum_{\alpha \in A} x_\alpha = L$. Let $S_n = \{\alpha \in A \mid x_\alpha > 1/n\}$. Then

$$L = \sum_{\alpha \in A} x_\alpha > \sum_{\alpha \in S_n} 1/n = \frac{|S_n|}{n}$$

So $| S_n| < nL$.

Let $S = \{\alpha \in A \mid x_\alpha > 0\}$. The $S$ is the countable union of each $S_n$, which are in turn each at most countable, so the result is at most countable.