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Let C(K) be the Banach space
of continuous, real-valued functions defined on a compact, Hausorff space, K,
let 𝒫 = 𝒫(K) be the positive linear forms, P, defined on C(K), for which
Pf ≤ supf(k) (k ∈ K), f ∈ C(K), and endow 𝒫(K) with the weak-star topology in
which it is, of course, compact. (As is well known, 𝒫 can be identified with the set of
countably additive probability measures defined on the Baire subsets of K.) Let
P∞ be the power probability on K∞, the product of denumerable number
of copies of K. Then, for each f ∈ C(K∞), the integral of f with respect
to P∞ is plainly continuous in P. As Theorem 2 below states, there are
no other continuous real-valued functions of P. The proof of this assertion
requires a generalization of Bernstein’s version of the celebrated polynomial
approximation theorem of Weierstrass, which generalization is provided by
Theorem 1.
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