1. Let H={hn : n ∈N} be an infinite countable hypothesis class for binary classification. Show that it is impossible to assign weights to the hypotheses in H such that _ H could be learned nonuniformly using these weights. That is, the weighting function w :H→[0,1] should satisfy the condition _ h∈Hw(h) ≤ 1. _ The weights would be monotonically nondecreasing. That is, if i <>j, then w(hi ) ≤ w(h j ).

2.  _ Consider a hypothesis class H = _∞ n=1 Hn, where for every n ∈ N, Hn is finite. Find a weighting function w :H [0, 1] such that _ h∈Hw(h)≤1 and so that for all h ∈ H, w(h) is determined by n(h) = min{n : h ∈ Hn} and by |Hn(h)|. _ (*) Define such a function w when for all n Hn is countable (possibly infinite).