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I'm having some trouble understanding Leo Breiman's proof that the generalization error of a random forest converges as the number of trees increases (here's a link to the paper).

At Appendix I he states that:

Proof of theorem 1.2: It suffices to show that there is a set of probability zero C on the sequence space $ \Theta_1, \Theta_2, ... $ such that outside of C, for all x, $$ \frac{1}{N}\sum_{n=1}^{N}I(h(\Theta_n,x)=j) \rightarrow P_\Theta(h(\Theta,x)=j) $$

I'm failing to understand the following:

  1. The sequence space of $ \Theta $'s is a sequence space as defined in wikipedia? are the elements of the sequence space infinite sequences of real or complex numbers?
  2. What does a "set of probability zero C on the sequence space ..." mean? is there another way to describe C?

Any help or explanation, partial or otherwise, is much welcomed.

Thanks !

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