Algorithmic randomness, reverse mathematics, and the dominated convergence theorem

Abstract

We analyze the pointwise convergence of a sequence of computable elements of L 1 (2 ω) 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 δ subset of Cantor space with positive measure has an element. This last statement is, in turn, equivalent to weak weak König'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 Σ 2 collection.

On the Uniform Computational Content of the Baire Category Theorem

“…The standard proof of the Lemma of Kučera relativizes directly to the formulation given above (see for instance the proof of [19, Lemma 6.10.1]). This leads to the following observation (see also [1,Theorem 3.7] for an account of the situation for 2-randomness in reverse mathematics). We recall that a point p ∈ 2 N is called (n + 1)-random for n ∈ N, if p ∈ MLR(∅ (n) ).…”

“…In particular, y is not a cluster point of (x n ) n and hence there is some s > 0 such that B(y, s) ⊆ B(x, r) and B(y, s) only contains finitely many x n . Since X is perfect, there is some z ∈ B(y, s 2 ) that is different from all these finitely many x n and hence positively bounded away from all x n , i.e., there is some t > 0 such that d(z, x n ) > t for all n ∈ N. This implies z ∈ B(x, r) \ A and since B(x, r) ⊆ B, this is a contradiction to (1). Hence, B(x, r) ⊆ A and hence x ∈ A • .…”

“…In case of the metric space of natural numbers N equipped with the discrete metric, one can consider a name p with respect to ψ − just as an enumeration of the complement of the represented set A. 1 By (A + (X), ψ + ) we denote the hyperspace A + (X) of closed subsets of a computable metric space X with respect to positive information. For a subset A ⊆ X of a topological space X we denote by A the closure of A.…”

On the computability of perfect subsets of sets with positive measure

“…Proof. It is obvious that over RCA 0 the instance of (3.1) for Π 0 2 sets implies 2−POS, which is the statement introduced by Avigad et al [1] that every Π 0 2 set with positive outer measure is non-empty. By [1, Theorem 3.7 ], RCA 0 + 2−POS ⊢ BΣ 0 2 .…”

“…A Π 0,X ′ 1 set is also a Π 0,X 2 set. By [1,Proposition 3.4], over RCA 0 + BΣ 0 2 every positive Π 0,X 2 set contains a positive Π 0,X ′ 1 set. Moreover, that every positive Π 0,X ′ 1 set contains a perfect subset where X ranges over all second order elements, trivially implies 2 − WWKL and thus also implies BΣ 0 2 by [1, Theorem 3.7].…”