  We prove convergence of a stochastic approximation of powers of the Riemann
$\zeta$ function to a non-Gaussian multiplicative chaos measure, and prove that
this measure is a non-trivial multifractal random measure. The results cover
both the subcritical and critical chaos. A basic ingredient of the proof is a
'good' Gaussian approximation of the induced random fields that is potentially
of independent interest.
