On Sep 8, 2015, at 7:48 PM, David Wilson <davidwwilson@comcast.net> wrote:

Is it possible to generalize measure on a set to non-real values, e.g. the surreal numbers?

Could this allow us to measure more sets, perhaps distinguish sets with real measure 0?

This is definitely doable, and has been done (at least by me). E.g., suppose you pick repeatedly, independently from the unit interval [0,1). Intuitively if you pick a high enough cardinal number K of times, one of those times should be (say) 1/3. Using surreal numbers, it indeed turns out that the intuitive calculation: Prob(not picking 1/3 in K picks) = (1-1/c)^K is equal to a positive infinitesimal. (Where c = the least surreal having the cardinality of the continuum, since the chance of missing 1/3 in a single pick is intuitively 1 - 1/c.) This can be made rigorous. —Dan