Let \(X\) and \(Y\) be topological spaces and let \(\Gamma\) be a family of subsets of \(X\). The standard pattern of continuity, i.e. a function \(f: X \to Y\) is \(\Gamma\)-continuous at a point \(x\in X\) iff for every neighborhood \(V\subseteq Y\) of \(f(x)\) there exists a neighbourhood \(U\subseteq X\) of \(x\) such that \(U \cap f^{-1}(V)\in \Gamma\), is considered for metric separable spaces. Topics from the quoted books are revised, concerning measurability with respect to Borel classes \(\Sigma^0_{\eta}\). Probably, the paper “Sur les définitions axiomatiques des ensembles mesurables (B)” by W. Sierpiński [Krak. Anz. 1918, 29–34 (1918; JFM 46.0295.01), Oeuvres choisies Tome II, PWN (1975; Zbl 0297.01029), 187–191] started this axiomatic attitude.
Methods which are appropriate for topological spaces with countable bases are applied to examine measurability. For example, a function \(f:X\to Y\) is \(\Sigma^0_{\xi}\) measurable when it is \(\Sigma^0_{\xi}\)-continuous, whenever \(X\), \(Y\) are metric separable spaces and \(\xi < \omega_1\). For real functions, a partial integral is introduced which coincides with the standard integral whenever the functions are \(\Gamma\)-continuous and \(\Gamma\) consists of measurable sets. The authors leave as open questions to find conditions under which the set of all points of \(\Gamma\)-continuity of a given real function is measurable; or to examine if classic theorems of the standard integral can be transfered to their partial integral.