popper-logic-scientific-discovery

appendix *iv 355
Although every Borel field of probabilities in Kolmogorov’s sense is
also one in our sense, the opposite is not the case. For we can construct
a system S 4 which is exactly like S 1, with h = (a, 1 2] still missing and
containing in its stead the open interval g = (a, 1 2), with p(g) = 1 2. We
define, somewhat arbitrarily, g¯ = u − g = ( 1 2, 1], and u − (g + g¯) = uū
(rather than the point 1 2). It is easily seen that S 4 is a Borel field in our
sense, with g as the product-element of A. But S 4 is not a Borel field in
Kolmogorov’s sense since it does not contain the set-theoretic product
of A: our definition allows an interpretation by a system of sets which is not a
Borel system of sets, and in which product and complement are not
exactly the set-theoretic product and complement. Thus our definition
is wider than Kolmogorov’s.
Our independence proofs of (i) and (ii) seem to me to shed some
light upon the functions performed by (i) and (ii). The function of (i)
is to exclude systems such as S 1, in order to ensure measure-theoretical
adequacy of the product (or limit) of a decreasing sequence: the limit
of the measures must be equal to the measure of the limit. The function
of (ii) is to exclude systems such as S 2, with increasing sequences
without limits. It is to ensure that every decreasing sequence has a
product in S and every increasing sequence a sum.