The reason those are all "f"s is that they are all versions of the same thing: the function mapping features to outputs, or approximations of it. The capital "F"s refer to random variables/processes describing the same function (using capitals for RVs and lower-case for samples is standard practice in statistics).
By using this notation he is drawing careful distinctions between the various approximations he's using. I think it's pretty good writing.
By using this notation he is drawing careful distinctions between the various approximations he's using. I think it's pretty good writing.