The extension of measurable functions

Abstract:

Say a measurable space $(Y,\mathcal {B})$ has the extension property (resp. the extension property in the restricted sense) if for every measurable space $(X,\mathcal {S})$ and every subset $A$ of $X$ (resp. subset $A$ of $X$ with $X\backslash A$ singleton), each function $f:A \to Y$ measurable for $\mathcal {S}(A) = \{ B \cap A:B \in \mathcal {S}\}$ may be extended to a measurable function $g:X \to Y$. A countably generated and separated $(Y,\mathcal {B})$ has the extension property if and only if it is a standard space, i.e. it is isomorphic to a Borel subset of the real line. The discrete space $(Y,{2^Y})$ has the extension property in the restricted sense if and only if the cardinality of $Y$ is not two-valued measurable.