Properties of Limit Inferior & Limit Superior

Lebesgue's Integrability Condition

Lebesgue's integrability condition, aka Lebesgue's criterion for Riemann integrability, Riemann--Lebesgue theorem

A bounded function on a compact interval [a, b] is Riemann integrable if and only if it is continuous a.e.

The criterion has nothing to do with the Lebesgue integral. It is due to Lebesgue and uses his measure zero, but makes use of neither Lebesgue's general measure or integral.

Almost Everywhere (a.e.) & Almost Surely (a.s.)

almost everywhere:  used in measure theory.
almost surely:  used in probability theory.