The percentile equals the integral of an indicator function, that some output variable, like loss, is greater than or less than some value. Take a percentile like the 90-th percentile on a test graded from 0 to 100. Say it is 85. So 10 percent of the class scored higher than 85. Suppose this was the true population distribution, not just the sample distribution. The indicator function for this is the score is greater than 85. The integral of this indicator function is .1, i.e. 10 percent.
We normally think of percentile a little bit differently, to set up this indicator function we already have to know the percentile. However, it is an easy way to see that the percentile corresponds to the integral of some indicator function. Here the integral is over the probability of sample outcomes, here the test score of an individual. We imagine we have an infinite population indexed by z, say and we integrate z over its range, the index set, which could be all real numbers from 0 to 1, corresponding to an uncountable population.