Pointwise measurable functions - Render, Herman; Rogge, Lothar (MATHEMATICA SCANDINAVICA, 2004)
We introduce the new concept of pointwise measurability. It is shown in this paper that a measurable function is measurable at each point and that for a large class of topological spaces the converse also holds. Moreover it can be seen that a function which is continuous at a point is Borel-measurable at this point too. Furthermore the set of measurability points is considered. If the range space is a $σ$-compact metric space, then this set is a $G_{δ}$-set; if the range space is only a Polish space this is in general not true any longer.