Vol. 45, No. 1, 1973

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Complementation problems for the Baire classes

William George Bade

Vol. 45 (1973), No. 1, 1–11

This paper initiates a study of the classes of Baire measurable functions on the unit interval I from the standpoint of the theory of spaces of continuous functions. For each countable ordinal α, the α-th Baire class Bα has a representation as Cα), where Ωα is a certain compactification of the discrete set I. For 1 α < β,Bα is a closed subalgebra of Bβ. The principal result proved here is the fact that Bα is always uncomplemented as a closed subspace of Bβ. The method of proof relies on a detailed analysis on the canonical onto map ϕ : Ωβ Ωα induced by the imbedding of Bα in Bβ, and consists of showing that this map admits no “averaging operator.” It depends heavily on recent results in the theory of averaging operators due to S. Z. Ditor.

Mathematical Subject Classification 2000
Primary: 46E25
Received: 18 October 1971
Published: 1 March 1973
William George Bade