On the Riemann zeta-function on the critical line (Q1066182)

Let \(C(R)\) be the space of continuous complex valued functions on the real line with the topology of uniform convergence on compact sets. The following problem is studied: does there exist a probability measure \(P\) defined for Borel sets \(A\) in \(C(R)\) such that the measure \[ (1/2T) \text{ mes}\{t\in [-T,T],\quad \zeta (1/2+ix+it)\in A\} \] converges weakly to \(P\) as \(T\) tends to infinity? Here \(\text{mes}\) means the Lebesgue measure. The author answers this question in the negative.