## Sunday, November 18, 2012

### Why infinitesimals are too small to help with infinite lotteries: Part IV

In previous parts, I argued that infinitesimals, or at least hyperreal infinitesimals, are too small to be the outcomes of a countably infinite lottery. Now it is time to extend this result to one hyperreal infinitesimal assignment in an uncountable lottery. Consider, then, the case of a uniform distribution on the interval [0,1) = { x:0≤x<1 }, say induced by a dart being thrown at a linear target. Bernstein and Wattenberg (1969) have shown that there is a hyperreal valued measure on all subsets of [0,1) such that (a) it is finitely additive, (b) it assigns infinitesimal probability to each singleton, and (c) it is almost translation invariant, in the sense that P(A@x) is within an infinitesimal of P(A), where A@x = {y@x : y in A}, and where y@x is addition modulo 1 (so, 0.5@0.7 = 0.2 and 0.2@0.3 = 0.5).

Now, just as in the standard construction of nonmeasurable sets, define the equivalence relation x~y on [0,1) by saying it holds if and only if there is a rational number q such that x@q=y. By the Axiom of Choice, let A0 be a set that contains exactly one representative from each equivalence class of [0,1) under ~. Let Q be all the rational numbers in [0,1) and let Aq=A0@q. Then the Aq are a partition of [0,1). If any one of them has non-infinitesimally positive measure, they all do, which violates finite additivity and total measure one. By (b) they must each have infinitesimal measure. But now we see that we can define a lottery on the countably infinite set Q by saying that q is the winner if and only if our uncountable lottery on [0,1) picked out some number in Aq. This lottery assigns an infinitesimal probability to each outcome in Q. But we have seen that a lottery that does that is a lottery that assigns far too small a value to each outcome. So we're still with the problem of infinitesimals being too small.