  We construct the analogue of Gaussian multiplicative chaos measures for the
local times of planar Brownian motion by exponentiating the square root of the
local times of small circles. We also consider a flat measure supported on
points whose local time is within a constant of the desired thickness level and
show a simple relation between the two objects. Our results extend those of
Bass, Burdzy and Khoshnevisan and in particular cover the entire $L^1$-phase or
subcritical regime. These results allow us to obtain a nondegenerate limit for
the appropriately rescaled size of thick points, thereby considerably refining
estimates of Dembo, Peres, Rosen and Zeitouni.
