Quote:
Originally Posted by CharlesL
Recently I have started reading the proof of the VC inequality in the appendix. On the bottom of page 190(Lemma A.3) , why does
sigma_S P[S] x P[sup_h E_in  E_in' > ... S ] <= sup_S P[sup_h E_in  E_in' >．．．S ]?
(Sorry for the terrible notations, I don't know how I can input math symbols)
what does it mean by taking the supremum on S?

All complicated math aside, supremum on S carries the physical meaning of taking the "maximum" value over all possible S.
So this inequality simply says an expected value (of P[sup_...]) is less than or equal to the maximum value.
Hope this helps.