The dominated convergence theorem implies that if is a sequence of functions on a probability space taking values in the interval , and converges pointwise a.e., then converges to the integral of the pointwise limit. Tao 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 on the rate of metastable convergence in the conclusion that is independent of the sequence and the underlying space. We prove a slight strengthening of Tao’s theorem which, moreover, provides an explicit description of the second bound in terms of the first. Specifically, we show that when the first bound is given by a continuous functional, the bound in the conclusion can be computed by a recursion along the tree of unsecured sequences. We also establish a quantitative version of Egorov’s theorem, and introduce a new mode of convergence related to these notions.
Algorithmic randomness, reverse mathematics, and the dominated convergence theorem, with J. Avigad and J. Rute. Annals of Pure and Applied Logic, Vol. 163, No. 12, pp. 1854-1864 (2012). arXiv: 1106.0775
We analyze the pointwise convergence of a sequence of computable elements of in terms of algorithmic randomness. We consider two ways of expressing the dominated convergence theorem and show that, over the base theory , each is equivalent to the assertion that every 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 collection.
We present a formal system, , which provides a faithful model of the proofs in Euclid’s Elements, including the use of diagrammatic reasoning.
Dedekind’s treatment of Galois theory in the Vorlesungen. Carnegie Mellon Univ., Dept. of Philosophy, Tech. Report 184.
We present a translation of §§160-166 of Dedekind’s Supplement XI to Dirchlet’s Vorlesungen über Zahlentheorie, which contain an investigation of the subfields of . In particular, Dedekind explores the lattice structure of these subfields, by studying isomorphisms between them. He also indicates how his ideas apply to Galois theory.
After a brief introduction, we summarize the translated excerpt, emphasizing its Galois-theoretic highlights. We then take issue with Kiernan’s characterization of Dedekind’s work in his extensive survey article on the history of Galois theory; Dedekind has a nearly complete realization of the modern “fundamental theorem of Galois theory” (for subfields of ), in stark contrast to the picture presented by Kiernan at points.
We translate a portion of Kronecker’s 1887 article “Über den Zahlbegriff,” in which he intends to demonstrate that all talk of algebraic numbers is unnecessary, in the sense that it can be eliminated in favor of (somewhat elaborate) talk of natural numbers.