  We prove that multiplicative chaos measures can be constructed from extreme
level sets or thick points of the underlying logarithmically correlated field.
We develop a method which covers the whole subcritical phase and only requires
asymptotics of suitable exponential moments for the field. As an application,
we establish these estimates hold for the logarithm of the absolute value of
the characteristic polynomial of a Haar distributed random unitary matrix
(CUE), using known asymptotics for Toeplitz determinant with (merging)
Fisher-Hartwig singularities. Hence, this proves a conjecture of Fyodorov and
Keating concerning the fluctuations of the volume of thick points of the CUE
characteristic polynomial.
