> I've been reading "Ten great ideas about chance", by Persi Diaconis.
> It includes a chapter which utilizes the idea of deriving
> probability from statistics, rather than the usual, other
> way round. Apparently, invented by Bruno de Finitti, whom
> the author idolizes.
> Anyhow, he discusses how means can imply probabilities.
Whittle, P. (2005). Probability via Expectation, 4th ed. Springer.
"We assume a sample space W, setting a level of description of the
realization of the system under study. In addition, we postulate that
to each numerical-valued observable X(w) can be attached a number E(X),
the expected value or expectation of X. The description of the variation
of w over W implied by the specification of these expectations will
be termed a probability process"
> And repeatedly refers to "the structure of probability", as explanation.
Pretty sure we are talking about Kolmogorov's "probability theory as
part of mathematics within the modern theory of measure and integral",
with "a real-valued random variable [being] a measurable function from
the basic set to the real numbers" [as above].
I enjoyed Charlie Geyer's reworking of Nelson 1987: