  We analyze the pointwise convergence of a sequence of computable elements of
L^1(2^omega) in terms of algorithmic randomness. We consider two ways of
expressing the dominated convergence theorem and show that, over the base
theory RCA_0, each is equivalent to the assertion that every G_delta subset of
Cantor space with positive measure has an element. This last statement is, in
turn, equivalent to weak weak K\"onig's lemma relativized to the Turing jump of
any set. It is also equivalent to the conjunction of the statement asserting
the existence of a 2-random relative to any given set and the principle of
Sigma_2 collection.
