A converse to Lebesgue's dominated convergence theorem

A metastable dominated convergence theorem

The dominated convergence theorem implies that if (fn) is a sequence of functions on a probability space taking values in the interval [0, 1], and (fn) converges pointwise a.e., then ( ∫ fn) converges to the integral of the pointwise limit. Tao [26] has proved a quantitative version of this theorem: given a uniform bound on the rates of metastable convergence in the hypothesis, there is a bound...

A Converse to Dye’s Theorem

Every non-amenable countable group induces orbit inequivalent ergodic equivalence relations on standard Borel probability spaces. Not every free, ergodic, measure preserving action of F2 on a standard Borel probability space is orbit equivalent to an action of a countable group on an inverse limit of finite spaces. There is a treeable non-hyperfinite Borel equivalence relation which is not univ...

Lebesgue's dominated convergence theorem in Bishop's style

We present a constructive proof in Bishop’s style of Lebesgue’s dominated convergence theorem in the abstract setting of ordered uniform spaces. The proof generalises to this setting a classical proof in the framework of uniform lattices presented by Hans Weber in “Uniform Lattices II: Order Continuity and Exhaustivity”, in Annali di Matematica Pura ed Applicata (IV), Vol. CLXV (1993). 1. Both ...

Ribet’s Converse to Herbrand’s Theorem

Algorithmic randomness, reverse mathematics, and the dominated convergence theorem

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