  The transition measure is a foundational concept introduced by Sergey Kerov
to represent the shape of a Young diagram as a centered probability measure on
the real line. Over a period of decades the transition measure turned out to be
an invaluable tool for many problems of the asymptotic representation theory of
the symmetric groups. Kerov also showed how to expand this notion for a wider
class of continual diagrams so that the transition measure provides a
homeomorphism between a subclass of continual diagrams (having a specific
support) and a class of centered probability measures with a support contained
in a specific interval. We quantify the modulus of continuity of this
homeomorphism. More specifically, we study the dependence of the cumulative
distribution function of Kerov transition measure on the profile of a diagram
at the locations where the profile is not too steep.
