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We prove that multiplicative chaos measures can be constructed from extremelevel sets or thick points of the underlying logarithmically correlated field.We develop a method which covers the whole subcritical phase and only requiresasymptotics of suitable exponential moments for the field. As an application,we establish these estimates hold for the logarithm of the absolute value ofthe 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 andKeating concerning the fluctuations of the volume of thick points of the CUEcharacteristic polynomial.

Multiplicative chaos measures from thick points of log-correlated fields