We analyze the pointwise convergence of a sequence of computable elements of
L^1(2^omega) in terms of algorithmic randomness. We then show that, over the
base theory RCA_0, a version of the dominated convergence theorem is equivalent
to the assertion that every G_delta subset of Cantor space with positive
measure has an element. It is also equivalent to a version of weak weak
K\"onig's lemma relativized to the Turing jump of any set. These principles
imply the existence of a 2-random relative to any given set, and are equivalent
to that assertion in the presence of Sigma^0_2 induction.