I was never compelled about valuation or the risk neutral. Here's something interesting.The difference between the n th squares 1 apart is 2n+1 so if you take sqrt(0.5(p^2- 1)-0.5(p^2+1)) for p prime you can get the solutions to pythagoras like 11^2 is 61^2-60^2 or 5^2,12+13 so pygathoras is really about the centre of p^2 and the 2 exponent relates to when you are going to cross. Obvious in retrospect. It is odd some of the binomial coefficients and how the number sets of size k has the set size 2 say or any subset of a fixed sized exactly the same frequency. So in a measure space you can have very different partitions can't you for your conditional expectation. Finally the probabability that a given given integer is an undecideable statement of the Godel type is order much bigger than the power set of those which can be shown to be true or false.