Discussion of the VC proof

CountVonCount · #24 10-26-2016, 03:28 PM

I have also another question on the same page (190):
At the end of the page there is the formula:

$\sum_S \mathbb{P}[S]\times\mathbb{P}[sup_{h\in H}\vert E_{in}(h) - {E_{in}}'(h)) \vert > \frac{\varepsilon }{2}] \leq sup_S \mathbb{P}[sup_{h\in H}\vert E_{in}(h) - {E_{in}}'(h)) \vert > \frac{\varepsilon }{2}] \vert S ]$

I don't understand why the RHS is greater or equal to the LHS. The only legitimation I see for this is, that the distribution of P[S] is uniform, but this has not been stated in the text.
Or do I oversee here anything and this is also valid for all kinds of distribution?

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