summable

Summable is an older, very evocative name for integrable, popular in the French tradition (sommable). The image is a bank ledger: you can add up all the contributions of f and get a finite total only if the gross flow — the integral of the absolute value — is itself finite. If the positive and negative parts both had infinite area, the books would never balance.

Formally, a measurable function f is summable if the integral of |f| dμ is finite; this is identical to being Lebesgue integrable and identical to lying in L^1. On a counting measure over the integers, summability of a sequence is exactly absolute summability of the corresponding series — the sum of |a_n| is finite — so the word reuses the same idea in the discrete case.

The connection to series is the clean way to remember why Lebesgue integrability is absolute. A series with the sum of |a_n| finite can be rearranged freely without changing its value; one that converges only conditionally cannot. Lebesgue integration deliberately admits only the absolute (summable) case, which is why it is so well-behaved under limits and rearrangements.