  The random trigonometric series $\sum_{n=1}^\infty \rho_n \cos (nt
+\omega_n)$ on the circle $\mathbb{T}$ are studied under the conditions $\sum
|\rho_n|^2=\infty$ and $\rho_n\to 0$, where $\{\omega_n\}$ are iid and
uniformly distributed on $\mathbb{T}$. They are almost surely not
Fourier-Stieljes series but define pseudo-functions. This leads us to develop
the theory of trigonometric multiplicative chaos, which are the limits of the
exponentiations of partials sums. which produces a class of random measures.
The behaviors of the partial sums of the above series are proved to be
multifractal. Our theory holds on the torus $\mathbb{T}^d$ of dimension $d\ge
1$.
