How to exploit knowledge of the sampling distribution for better generalization bounds?

Question

In the PAC learning model, suppose the learner actually knows the sampling distribution $P$. Surely this knowledge can be exploited to yield better generalization bounds -- but how? One idea is using PAC-Bayesian techniques to convert the distribution over $X$ into a distribution over hypotheses, which is what I think this paper is doing: http://www0.cs.ucl.ac.uk/staff/G.Lever/pubs/ALTfinal.pdf

Any other methods for making clever use of $P$?

[Edit 25.03.2015: In light of Vitaly's answer, let me sharpen the question. Suppose that I know the sampling distribution only approximately as $Q$. Can we formulate a generalization bound in terms of $D(Q||P)$ or $||P=Q||_{TV}$?]

Answer 1

For specific distribution over $X$ uniform convergence bounds scale linearly with the logarithm of the size of the (smallest) $\epsilon$-cover of the concept class $C$ (the cover can use any functions over $X$). This is proved in "Learnability with respect to fixed distributions" paper of Benedek and Itai. As a result sample complexity of PAC learning scales linearly with this parameter.