The measurable boundaries of a real function (Q1820260)

In the beginning [Acta Math. 65, 263-282 (1935; Zbl 0013.15302)] \textit{H. Blumberg} defined the notions of upper and lower measurable boundaries of a real function and used these entities to obtain several interesting results that pertain to arbitrary real functions. These results have a rather narrow range of applicability, since the definitions and the methods employed by Blumberg use such special devices as density theorems, approximate continuity, and the Vitali covering theorem. Subsequently \textit{C. Goffman} and the reviewer [Fundam. Math. 48, 105-111 (1960; Zbl 0095.044)] presented a definition of measurable boundary that makes sense in any totally-\(\sigma\)-finite measure space. The latter definition agrees with the former, for Lebesgue measure on the real line, and, in the general situation, yields proofs that are simpler than those given by Blumberg for the special case. The later paper is built upon an ancient result according to which the measure ring engendered by a totally-\(\sigma\)-finite measure space \((X,{\mathcal S},\mu)\) is a complete lattice, from which it follows that the lattice of equivalence classes of extended-real-valued measurable functions on X is also complete. The present work is carried on within the framework of a measurable space \((X,{\mathcal S})\) and a proper \(\sigma\)-ideal \({\mathcal T}\) of \({\mathcal S}\). In this setting, two measurable functions f and g are equivalent if and only if \(\{x:f(x)\neq g(x)\}\) belongs to \({\mathcal T}\). Denoting by \({\mathcal M}\) the set of classes of measurable functions equivalent in this sense, it follows that both \({\mathcal S}/{\mathcal T}\) and \({\mathcal M}\) are complete lattices, if \({\mathcal S}/{\mathcal T}\) satisfies the countable chain condition. The completeness of \({\mathcal M}\) enables the author to define the measurable boundaries of an extended-real-valued function on X in a manner completely analogous to that employed when \({\mathcal T}\) is the null-set ideal in a totally-\(\sigma\)-finite measure space, and to establish appropriate generalizations of the theorem of Saks and Sierpiński analogous to those given in the article cited above.