Remarks on measurable function spaces (Q1098059)

Let X be a metrizable space. The space of measurable functions from [0,1] into X is denoted by M(X), and the subspace \(\{\) \(f\in M(X):\) f is a `simple function' having constant values on sections of a subdivision of [0,1]\(\}\) is denoted by \(M_ S(X)\). If \(0<\beta <1\) and \(\pi\) represents the partition \(0=a_ 0<a_ 1<a_ 2<...<a_ m=1\), then \(\beta_{\pi}(f,g)\) is defined to be f(t) if \(a_ j<t\leq (1-\beta)a_ j+a_{j+1}\), g(t) if \((1-\beta)a_ j+a_{j+1}<t\leq a_{j+1}\), \(j=1,2,...,m\), where f,g are in M(X). A subset \(D\subseteq M(X)\) is said to be convex if f,g in D implies that \(\beta_{\pi}(f,g)\in D\) for all partitions of [0,1] and all numbers \(0<\beta <1.\) Following terminology introduced in, for example, \textit{C. Besaga} and \textit{A. Pełczyński}, ``Selected topics in infinite dimensional topology'' (1975; Zbl 0304.57001), the results of this paper indicate that (i) If \(D\subseteq M(X)\) and there is a dense subset C of D consisting of simple functions, then D is absolutely retract; (ii) if X is a separable compact metrizable space having more than one point and \(X_ 0\subseteq X\) is a countable dense subset, then \((M(X),M_ S(X_ 0))\simeq (\ell_ 2,\ell\) \(f_ 2)\), where \(\ell\) \(f_ 2\) is a specified subset of \(\ell_ 2.\) The results form a sequel to the author's earlier papers including Topology Appl. 22, 109-122 (1986; Zbl 0613.54002).