On measurable orthogonally exponential functions

Abstract.

Let X be a real linear topological space and \(f:X\rightarrow {\Bbb C}\) satisfy \(f(x+y)=f(x)f(y)\) whenever \(x\perp y\), where \(\perp \subset X^2\) is defined by (01) – (04). We show that if f is universally, Christensen, or Baire measurable (under suitable assumptions on X), then \(f(x)=\exp (A_1(x)+iA_2(x)+cL(x,x))\) for \(x\in X\) with some continuous linear functionals \(A_1,A_2:X\rightarrow {\Bbb R}\), bilinear \(L:X^2\rightarrow {\Bbb R}\), and \(c\in {\Bbb C}\).