On continuity of measurable cocycles (Q2706917)

Let \(X\) be a compact metric space and \(K\) be the set of all real or complex numbers. Given a function \(F : ]0,\infty[\times X\to X\), a solution \(G : ]0,\infty[\times X\to K\) of the equation NEWLINE\[NEWLINEG(s + t, x) =G(s,x)G(t,F(s, x))NEWLINE\]NEWLINE is called a cocycle of \(F\) (or simply a cocycle) if for every \(t\in ]0,\infty[\), \(G(t,\cdot)\) is continuous. A cocycle \(G\) is said to be measurable if for every \(x\in X\), \(G(\cdot,x)\) is Lebesgue measurable. The author, among others, proves the following theorem:NEWLINENEWLINENEWLINELet \(G : [0,\infty]\times X\to K\) be a measurable non-vanishing cocycle. Then the mapping \(t\mapsto G(t,\cdot)\in{\mathcal C}(X,K)\) is continuous where \(t\in]0,\infty[\). In particular, the function \(G\) is continuous.