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Algorithmic randomness, reverse mathematics, and the dominated convergence theorem. (English) Zbl 1259.03021
The authors use techniques of reverse mathematics to analyze the relative strength of several versions of the dominated convergence theorem for Lebesgue integration. Let DCT\(^\prime\) denote the assertion that, given \(f\), \(g\), and a sequence \(\langle f_n \rangle\) of elements of \(\mathcal L^1 (X)\), if \(\langle f_n \rangle\) is dominated by \(g\) and converges pointwise a.e. to \(f\), then \(\langle \int f_n \rangle\) converges to \(\int f\). Working in RCA\(_0\), the authors prove that DCT\(^\prime\) is equivalent to DCT\(^\ast\), which says that if \(\langle f_n (x) \rangle\) is Cauchy a.e.  and dominated by \(g\), then \(\langle \int f_n \rangle\) is Cauchy also. Both of these versions of the dominated convergence theorem are shown to be equivalent to the principle 2-POS, which asserts that any \(G_\delta\) subset of Cantor space with positive measure is nonempty. The principle 2-POS is equivalent to B\(\Sigma^0_2\) plus 2-RAN, that is, to a pigeonhole principle plus a formalization of the existence of 2-random sets. All these principles are stronger than WWKL, incomparable to WKL, and strictly weaker than ACA, refuting a conjecture of Simpson related to the stronger form of DCT analyzed by X. Yu [Math. Log. Q. 40, No. 1, 1–13 (1994; Zbl 0804.03047)].

MSC:
03B30 Foundations of classical theories (including reverse mathematics)
03F35 Second- and higher-order arithmetic and fragments
03D32 Algorithmic randomness and dimension
03F60 Constructive and recursive analysis
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