The definition of h!k makes sense for anyh, that is, the superscript !k in this context is a mapping from finite histories to sets of pairs as you said. In the line in question we just apply this mapping to x:n where x is a bound variable coming from the expected value.
I don’t believe that x!k:n was defined anywhere, but we “use the definition” in the proof of Lemma 1.
As far as I can tell, it’s a set of (j,y) pairs, where j is the index of a hypothesis, and y is an infinite history string, rather like the set h!k.
How do the definitions of h!k and x!k:n differ?
Hi Alex!
The definition of h!k makes sense for any h, that is, the superscript !k in this context is a mapping from finite histories to sets of pairs as you said. In the line in question we just apply this mapping to x:n where x is a bound variable coming from the expected value.
I hope this helps?