popper-logic-scientific-discovery

appendix *iv 353
(ii) An admissible system S is called a Borel field of probabilities if, and only
if, there is in S a product-element of any (absolutely or relatively)
decreasing sequence of elements of S.
Of these two definitions, (i) corresponds precisely to Kolmogorov’s
so-called ‘axiom of continuity’, while (ii) plays a part in our
system analogous to Kolmogorov’s definition of Borel fields of
probability.
It can now be shown that whenever S is a Borel field of probabilities in
Kolmogorov’s sense, it is also one in the sense here defined, with probability as a countably
additive measure function of the sets which are the elements of S.
The definitions of admissible systems and Borel fields of probabilities
are framed in such a way that all systems S satisfying our
postulates and containing only a finite number of different elements
are admissible systems and Borel fields; accordingly, our definitions are
interesting only in connection with systems S containing an infinite number
of different elements. Such infinite systems may, or may not, satisfy the one
or the other or both of our defining conditions; in other words, for
infinite systems our defining conditions are non-redundant or
independent.
This non-redundancy can be proved for (i) most easily in that form
of it which is mentioned in footnote 12, with the help of the example
of the missing half-interval, S1, given above. All we have to do is to
define probability p(x) as equal to l(x), that is to say, the length of the
interval x. Our first definition, (i), is then violated since lim p(an) = 1 2
while for the product-element (in S) of A, p(a) = 0. Definition (ii) is
violated by our example S2 (which vacuously satisfies the first
definition).
While the first of these examples establishes the independence or
more precisely the non-redundancy of our first definition—by violating
it—it does not, as it stands, establish the independence of
Kolmogorov’s ‘axiom of continuity’ which is clearly satisfied by our
example. For the missing half-interval, h = (0, 1
2], whether in S or not, is
the only set-theoretic product of A, so that a = h is true for the settheorist,
whether or not a is in S. And with a = h, we have
lim p(an) = p(a). Thus Kolmogorov’s axiom is satisfied (even if we omit
the condition p(ā, a) ≠ 0; cf. footnote 12).