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On the space of continuous functions on a dyadic set. (English) Zbl 0776.46019

Summary: In the part (16-3) of his extensive study on measurability in Banach spaces, M. Talagrand [Pettis integral and measure theory, Mem. Am. Math. Soc. 307, 224 p. (1984; Zbl 0582.46049)] considered the Banach space \(C(K)\) of continuous functions on a dyadic topological space \(K\). He proved that \(C(K)\) is realcompact in its weak topology, if, and only if, the topological weight of \(K\) is not a two-measurable cardinal (Theorem 16-3-1). Then he asked for an alternative to a rather complicated proof presented there (p. 214) and posed the problem whether \(C(K)\) is measure-compact whenever the weight of \(K\) is not a real- measurable cardinal (Problem 16-3-2).
It turns out that measure-theoretic properties of the weak topology on \(C(K)\) are closely connected with the existence of the so-called sequential cardinals, discussed earlier by Mazur, Noble, Antonowskij- Chudnovsky and Ciesielski. We offer a new proof of Talagrand’s theorem mentioned above, based on our result on continuity of functions on a Cantor cube, which in fact is a variation of the result from [M. Ya. Antonovskij and D. V. Chudnovskij, Russ. Math. Surveys 31, No. 3, 69-128 (1976; Zbl 0355.54006)]. Moreover we show that \(C(K)\) is measure- compact if the weight of \(K\) is not a sequential cardinal. This answers Talagrand’s problem affirmatively provided Martin’s axiom holds.

MSC:

46E15 Banach spaces of continuous, differentiable or analytic functions
46G10 Vector-valued measures and integration
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