Is there a article/textbook that treats probability distributions on functions just like basic textbooks cover the classical distributions for scalar variables?
Suppose the random function $f$ has a gaussian process distribution with some mean function $m$ and covariance function $w$. I know how to generate random (functional) deviates from that distribution.
What I'd like to know, for example, is given another function $g$ (that can be evaluated at an arbitrary number of points over the same range than where $f$ is defined), what is the probability that $g$ or a more extreme function was drawn from $f$'s distribution. There must be a way to define a cumulative distribution function and a probability density function for functions!