  We study Mandelbrot's multiplicative cascade measures at the critical
temperature. As has been recently shown by Barral, Rhodes and Vargas
(arXiv:1203.5445), an appropriately normalized sequence of cascade measures
converges weakly in probability to a nontrivial limit measure. We prove that
these limit measures have no atoms and give bounds for the modulus of
continuity of the cumulative distribution of the measure. Using the earlier
work of Barral and Seuret (2007), we compute the multifractal spectrum of the
measures. We also extend the result of Benjamini and Schramm (2009), in which
the KPZ formula from quantum gravity is validated for the high temperature
cascade measures, to the critical and low temperature cases.
