Isometries of measure algebras (Q788138)

\textit{P. Gruber} [Isr. J. Math. 42, 277-283 (1982; Zbl 0502.52006)) asked for a characterization of the isometries of the following metric space (\({\mathfrak E},d)\). Let (X,\({\mathcal B},\mu)\) be a measure space, \({\mathfrak A}={\mathcal B}/\mu\) its measure algebra and \({\mathfrak E}\) the set of elements with finite measure in \({\mathfrak A}\). The metric d is the Nikodym metric defined by \(d(a,b)=\mu(a\Delta b)\). We show that, for a \(\sigma\)-finite Borel measure \(\mu\) on a Polish space X, the isometries T of \({\mathfrak E}\) are exactly of the form \[ (^*)\quad T([A])=[\phi(A)\Delta A_ 0], \] where \(\phi\) is a measure preserving Borel isomorphism and \(A_ 0\) a fixed set of finite measure. For X Polish and \(\mu\) an atomless Radon measure we also consider the subspaces \({\mathfrak F}_ e\) and \({\mathfrak K}\) of \({\mathfrak E}\) consisting of the equivalence classes of all closed and all compact sets, respectively. We show that the isometries of these spaces are also of type \((^*)\) where, in addition, \(\phi\) is an almost homeomorphism - further satisfying a compactness condition in the case of \({\mathfrak K}\). By an almost homeomorphism we mean a Borel isomorphism which is a homeomorphism on an invariant set whose complement has measure zero.