Weak measure extension axioms

We consider axioms asserting that Lebesgue measure on the real line may be extended to measure a few new nonmeasurable sets. Strong versions of such axioms, such as real-valued measurability, involve large cardinals, but weak versions do not. We discuss weak versions which are sufficient to prove various combinatorial results, such as the nonexistence of Ramsey ultrafilters, the existence of ccc spaces whose product is not ccc, and the existence of S- and L-spaces. We also prove an absoluteness theorem stating that assuming our axiom, every sentence of an appropriate logical form which is forced to be true in the random real extension of the universe is in fact already true.