Are there conditions which we can impose on a set of Riemann Integrable Functions that are weaker than equi-continuity condition of the original Arzela-Ascoli theorem which still have the result of the theorem hold (as in, the set being compact)?
The first obvious problem is that an arbitrary set of Riemann integrable functions may not be closed under pointwise limits(though the set of Riemann integrable functions is closed under uniform limits), so this weaker condition must restrict the set to one that guarantees any limit of functions must be Riemann integrable.