Consider the heirarchy of sets(adsbygoogle = window.adsbygoogle || []).push({});

finite sets

N

P(N)

P(P(N))

...

And let V denote the class of all sets equipollent to a set in that list.

I want to turn V into a psuedo-probability space and I need help. I say pseudo because the underlying space isn't a set.

Let A be in V. Define o(A) to be 0 if A is finite and otherwise, o(A)=n where n is the number of times the power set was applied to N to get to a set on the above list which A is equipollent to.

Example, R is in V. R~P(N), so here n=1. Then o(R)=1. N is also in V and there, n=0; so o(N)=0.

o is an outer measure because:

1. o(Ø)=0 and

2. o(AuB)<=o(A)+o(B). To see this, first consider the case where AuB is finite. Then 0 <= anything, so we're done. Now consider the case when N<=AuB where that means there is a 1-1 map from N to AuB. Suppose A~P(...P(N)) where there are n_A power set operations and similarly there is a n_B for B. Then that index for AuB is max(n_A, n_B). Let's denote this by q. o(AuB)=q. o(A)=n_A and o(B)=n_B. It's pretty clear now that o(AuB)<=o(A)+o(B) because q equals either n_A or n_B.

I think the proof would still work for a countable collection (ie sigma-subadditivity), albiet some things would have to be changed.

The question is how can I construct from this an m that takes elements in V to [0,oo] such that sub-additivity is replaced by additivity. That is, for a pair of disjoint sets in V, m(AuB)=m(A)+m(B).

m will be some nontrivial function of o.

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# Measure from outer measure

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