On Radon measures on first-countable spaces

It is shown that every Radon measure on a first-countable Hausdorff space is separable provided ω1 is a precaliber of every measurable algebra. As the latter is implied by MA(ω1), the result answers a problem due to D. H. Fremlin. Answering the problem posed by D. H. Fremlin ([4], 32R(c)), we show in this note that, assuming (∗) ω1 is a precaliber of every measurable Boolean algebra, every Radon measure on a first-countable space is separable. We treat here only finite measures. By the Maharam type of a measure μ we mean the density character of the Banach space L1(μ) (see [4] or [5]). Thus the Maharam type of μ is the least cardinal κ for which there exists a family D of measurable sets such that |D| = κ, and D approximates all measurable sets, that is, for every measurable B and ε > 0 there is D ∈ D with μ(B4D) < ε. In particular, a measure μ of Maharam type ω is called separable. Basic facts concerning Radon measures can be found in [7] or [5]. Although one can use several definitions of a Radon measure, differences are not so important when the measure in question is finite. Let us agree that, given a topological space S, the statement “μ is a Radon measure on S” means that μ is defined on some σ-algebra containing all open subsets of S, and μ(B) = sup{μ(K) : K ⊆ B, K compact} for every measurable set B. Recall that ω1 is said to be a precaliber of a Boolean algebra A if for every family {aξ : ξ < ω1} of non-zero elements of A one can find an uncountable set X ⊆ ω1 such that the family {aξ : ξ ∈ X} is centered, that is, ∏ ξ∈I aξ 6= 0 for every finite I ⊆ X (see [6], A2T). Recall also that a 1991 Mathematics Subject Classification: Primary 28C15; Secondary 54A25. Partially supported by KBN grant 2 P 301 043 07.