Algorithmic randomness, reverse mathematics, and the dominated convergence theorem

We analyze the pointwise convergence of a sequence of computable elements ofL¹(2^ω) in terms of algorithmic randomness. We consider two ways of expressing the dominated convergence theorem and show that, over the base theory RCA₀, each is equivalent to the assertion that every G_δ subset of Cantor space with positive measure has an element. This last statement is, in turn, equivalent to weak weak Königʼs 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 Σ₂ collection.