Christensen measurable solutions of generalized Cauchy functional equations (Q1096814)

The relation between measurability and continuity of solutions of various functional equations has been studied by many authors. The author of this paper studies similar problems. Let \((G,+)\) be an Abelian Polish topological group and \({\mathfrak M}\) the \(\sigma\)-algebra of all universally measurable subsets of G. A set \(A\in {\mathfrak M}\) is a Haar zero set if there exists a probability measure \(\mu\) on \({\mathfrak M}\) such that \(\mu (A+x)=0\) for each \(x\in G\). All subsets of Haar zero sets form a proper \(\sigma\)-ideal \({\mathcal C}_ 0\) of Christensen zero sets. The \(\sigma\)- algebra consisting of all sets of the form \(A=B\cup C\), where \(B\in {\mathfrak M}\) and \(C\in {\mathfrak C}_ 0\) is denoted by \({\mathcal C}\). Elements of the \(\sigma\)-algebra \({\mathcal C}\) are called Christensen measurable sets. Let X be a topological space, the function \(f: G\to X\) is Christensen measurable if \(f^{-1}(U)\in {\mathcal C}\) for every open subset U of X. Let \(X_ 1,...,X_ n\) be topological spaces with countable bases. The main result is the following: Suppose that \(H: X_ 1\times...\times X_ n\to X\) and \(g_ i:\) \(G\to X_ i\) is a Christensen measurable function for \(i=1,...,n\), then there exists a set \(M\subset G\), \(M\in {\mathcal C}/{\mathcal C}_ 0\) such that the restriction of H to \(g_ 1(M)\times...\times g_ n(M)\) is continuous. For a given neighbourhood W of zero in G let \(c: W\to G\) be a function continuous at zero such that \(c(0)=0\). If \(k_ 1,...,k_ n\) are fixed integers and \(g: W\to X\) satisfies the equation \(g(y)=H(g_ 1(x+k_ 1c(y)),...,g_ n(x+k_ nc(y))),\) \(x\in M\), \(y\in W\), then g is continuous at zero. The author studies the equation \(f(x+y)=H(f(x),f(y)),\) \(x,y\in G\) and shows that under certain restrictive hypotheses on the equation the Christensen measurability of f implies continuity of f without the usual assumption that H itself is continuous.