Question

when is f integrable?
in other words, f is integrable iff?

Answer 1

l.) Let f be a bounded function from an interval
[a, b] to R. We say that f is integrable on [a, b] if there is a number V such
that for every sequence of partitions {Pn} on [a, b] such that {µ(Pn)} → 0,
and every sequence {Sn} where Sn is a sample for Pn
{
X(f, Pn, Sn)} → V.