A localization of Γ-measurability

We introduce the notion of Γ continuity at a point x—where Γ is any pointclass—and give conditions under which Γ continuity at every x is equivalent to Γ measurability. Using this we extend the notion of the integral of a measurable function. Also we examine the case Γ = Σ ξ 0 , where ( Σ ξ 0 ) ξ < ω 1 is the usual ramification of the class of Borel sets, see [A.S. Kechris, Classical Descriptive Set Theory, Springer-Verlag, 1994].