I've seen this sort of integral come up when I've been studying approximating step processes to some very simple stochastic processes, but I can't shake the feeling that they link to some more general (and possibly, statistical or analytical?) idea that I am failing to recall the name of.
What is the name, if any, for integrals of the form $\frac{1}{b-a}\int_a^b f(x) \ \mathrm{dx}$? Is there any such name for the integrals of step functions? For example, $\frac{1}{x_1-x_0}\int_{x_0}^{x_1} c_1 \ \mathrm{dx}$?
When we wish to take the (arithmetic) mean of finitely values, we add them up and divide by the number of values. Analogously, we traditionally define the mean of a function $f$ on an interval $[a,b]$ to be given by an expression like the one in the question: $\dfrac{1}{b-a}\int_a^bf(x)\,\mathrm dx$. This terminology comes up in the name of the mean value theorem for integrals.
As an aside, do not confuse with the calculation of expected value ("mean") given a discrete or continuous probability distribution. That sort of calculation is more like "the mean of all possible rolls of a die, weighted by probability" than "the mean of these particular rolls of a die that happened".