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Non-cupping, measure and computably enumerable splittings

We show that there is a computably enumerable function f (that is, computably approximable from below) that dominates almost all functions, and f ⊕ W is incomplete for all incomplete computably enumerable sets W. Our main methodology is the LR equivalence relation on reals: A ≡LRB if and only if the notions of A-randomness and B-randomness coincide. We also show that there are c.e. sets that cannot be split into two c.e. sets of the same LR degree. Moreover, a c.e. set is low for random if and only if it computes no c.e. set with this property.