On local times for functions and stochastic processes .1.

Let X(t), 0 less than or equal to t less than or equal to 1, be a real-valued measurable function having a local time alpha(t, u), 0 less than or equal to t less than or equal to 1, u epsilon R. If the latter is continuous in t for a.e. u, then the distribution F(t, x) = integral(R) 1(alpha(t, u) > x) du and the monotone rearrangement alpha*(t, u) = inf{x: F(t, x) < u} of the local time alpha(t, u) are the local times for xi(s) = alpha(s, X(s)) and xi*(s) = F(s, X(s)), 0 less than or equal to s less than or equal to 1, respectively.