  We study one-dimensional exact scaling lognormal multiplicative chaos
measures at criticality. Our main results are the determination of the exact
asymptotics of the right tail of the distribution of the total mass of the
measure, and an almost sure upper bound for the modulus of continuity of the
cumulative distribution function of the measure. We also find an almost sure
lower bound for the increments of the measure almost everywhere with respect to
the measure itself, strong enough to show that the measure is supported on a
set of Hausdorff dimension $0$.
