Lemma The Archimedean Probability Lemma [efr-ADS6]

Let \mathcal {C} be a distributive coinflip Markov category where the Kolmogorov product 2^\omega exists, let \chi : 2^\omega \to A be a deterministic map. Suppose there exist maps d: A \otimes A \to A and s: 2^\omega \to 2^\omega satisfying:

  1. For every finite N, \pi _{2^N}s : 2^\omega \to 2^\omega \to 2^N factors over some finite projection 2^\omega \to 2^M. Moreover each of these factorizations 2^M \to 2^N is given by a map of finite sets which has fibers of uniform size.
  2. \chi factors over every cofinite projection 2^\omega \to 2^{\omega _{>N}}.
  3. d(\chi (x), \chi (s(x))) = 1 as maps 2^\omega \to 2.
  4. d(x,x) = x.
Then \chi \circ f^\omega = 1, where f^\omega : I \to 2^\omega is the independent pairing of \omega -many coinflips.