Spaces that are close to absolute retracts (Q1077753)

In his earlier paper the author introduced a natural generalization of the notions of retract and absolute retract. Those generalized notions can be defined as follows: Let A be a closed subset of a compact metric space X. (1) Let \(\mu\) be a Radon measure on A. The set A is a \(\mu\)- almost retract of X whenever for every \(\epsilon >0\) there exist a compact \(A_{\epsilon}\subset A\) and a map \(r_{\epsilon}: X\to A\) such that \(\mu (A-A_{\epsilon})<\epsilon\) and \(r_{\epsilon}| A_{\epsilon}=id_{A_{\epsilon}}\). (2) The set A is a t-retract of X whenever A is a \(\mu\)-almost retract of X for every Radon measure \(\mu\) on A. (3) The class \(AR_ t\) of absolute t-retracts is defined in the obvious way. \(\{\) Reviewer's remark: It is hard to guess what is the origin of the term ''t-retract''\(\}\). In the present paper the author extends the notions of t-retract and \(AR_ t\) to some (less natural) notions of \(t^*\)-retract and \(AR_{t^*}\). He proves a theorem which could be (more clearly) formulated in terms of the following natural notion of \(\mu\)-almost extendability: A \(\mu\)-measurable map f:A\(\to Y\) is \(\mu\)-almost extendable over X whenever for every \(\epsilon >0\) there exist a compact \(A_{\epsilon}\subset A\) and a map \(f_{\epsilon}: X\to Y\) such that \(\mu (A-A_{\epsilon})<\epsilon\) and \(f_{\epsilon}| A_{\epsilon}=f| A_{\epsilon}.\) Then Theorem 1 claims what follows: If X is a binormal space, A its compact subset, \(\mu\) a Radon measure on A, and \(Y_ 0\in ANR\) is homotopically dominated by \(Y\in AR_{t^*}\), then every \(\mu\)- measurable map \(f: A\to Y_ 0\) is \(\mu\)-almost extendable over X.