On strong liftings for projective limits

We discuss the permanence of strong liftings under the formation of projective limits. The results are based on an appropriate consistency condition of the liftings with the projective system called “self-consistency”, which is fulfilled in many situations. In addition, we study the relationship of self-consistency and completion regularity as well as projective limits of lifting topologies. Introduction. Only recently in [19] the general theory of inductive limits of (topological) measure spaces was developed by N. D. Macheras and at the same time the permanence of strong lifting was established for inductive limits. For much longer time the projective limit of measure spaces is in common use (see e.g. Bochner [4], Choksi [5], Musia l [25], Rao [29]) but there seems to be no discussion of the permanence of strong liftings for general projective limits. Only for finite or countable products there exist permanence results in the forthcoming paper [22]. As in [22] our main concern in this paper is with conditions of compatibility of the liftings in the factors with the lifting on the limit or product. The notion of the “consistent lifting” of M. Talagrand [30] seems to be the first example on this line. Talagrand’s paper is only for finite products in which all factors must be equal. [22] gives consistency conditions for finite and countable products with different factors, and our basic sufficient condition for the existence of strong liftings on projective limits in this paper, the socalled “self-consistency” (see Section 1 for definitions), may be read as a strengthening of Talagrand’s consistent lifting, i.e. to be precise in his special instance it is a condition in terms of the generators of the product σ-algebra while ours is a condition on the whole σ-algebra. Remark 2.2(iii) gives a list of projective systems which allow selfconsistent liftings. Among them are always countable systems provided all factors have the universally strong lifting property (USLP for short). The basic existence result for 1991 Mathematics Subject Classification: Primary 28A51; Secondary 60B05. 210 N. D. Macheras and W. Strauss strong liftings on projective limits is Theorem 2.3. It can be seen that selfconsistency is not a necessary condition for the existence of strong liftings (see Remark 3.3). For products there is a well-known but somewhat elusive relationship between completion regularity and the existence of strong liftings (see e.g. [6]). By Theorem 3.1, self-consistency is sufficient (but again not necessary by Remark 3.3) for the permanence of completion regularity under projective limits of compact spaces and more generally for Hausdorff completely regular spaces in the presence of sequential maximality. Corollary 3.2 gives a permanence result for strong Baire liftings in projective limits. In Section 4 we study the projective limit for lifting topologies. In terms of lifting topologies equivalent conditions can be given for the existence of a strong lifting on the projective limit (see Theorem 4.1 and its corollaries). Section 5 contains the specialization to products, thus extending the results in [22] from countable to uncountable products (see Theorem 5.3). One consequence of the basic result (Theorem 5.3) is the existence of a strong lifting if all factors have the USLP (see Theorem 5.6). This theorem comprises the classical result of [14] and [16]. There exist projective limit Radon measures, e.g. the Wiener measure on I where I := [−∞,+∞] in which every lifting is almost strong, i.e. they have the USLP but there exists no strong lifting which can be represented as a projective limit of strong liftings. 1. Preliminaries. We assume throughout that every topological space X is Hausdorff completely regular. The σ-field of Borel (resp. Baire) sets over X, B(X) (resp. B0(X)), is the one generated by all open subsets of X (resp. by all bounded continuous functions on X). By a Borel (resp. Baire) measure on X we mean a finite, nonnegative countably additive set function defined on B(X) (resp. B0(X)). Let (Ω,Σ, μ) be a finite measure space, i.e. Ω is a set, Σ a σ-field of subsets of Ω, and μ a nonnegative real-valued countably additive measure on Σ. Throughout we assume that 0 < μ(Ω) < ∞. We write Σ∧ for the Carathéodory completion of Σ with respect to μ. The canonical extension of μ to Σ∧ will again be denoted by μ. Let (T,Λ) be a measurable space, i.e. T is a set and Λ a σ-field of subsets of T . A mapping f from Ω into T is called Σ-Λmeasurable iff f−1(B) ∈ Σ for all B ∈ Λ. L∞(Ω,Σ, μ) or just L∞(Ω,μ) is the space of all bounded ΣB(R)-measurable functions on Ω, where R denotes the set of all real numbers. For a complete finite measure space (Ω,Σ, μ) a lifting on L∞(Ω,μ) is a linear mapping %∗ from L∞(Ω,μ) into L∞(Ω,μ) with the following properties: Strong liftings for projective limits 211 (I) %∗(f) = f a.e. (μ), (II) f = g a.e. (μ) implies %∗(f) = %∗(g), (III) %∗(1) = 1 where 1 is the function identically equal to 1 on Ω, (IV) f ≥ 0 a.e. (μ) implies %∗(f) ≥ 0, (V) %∗(fg) = %∗(f)%∗(g) (cf. [14, p. 34]). A lifting on Σ is a mapping % from Σ into Σ with the following properties: (I′) %(A) = A a.e. (μ), (II′) A = B a.e. (μ) implies %(A) = %(B), (III′) %(Ω) = Ω, %(∅) = ∅, (IV′) %(A ∩B) = %(A) ∩ %(B), (V′) %(A ∪B) = %(A) ∪ %(B) (cf. [14, p. 35]). A mapping φ from Σ into Σ is called a lower density (or just a density) for (Ω,Σ, μ) if it satisfies (I′)–(IV′) (cf. [14, p. 36]). We note that for any lifting % on Σ there exists exactly one lifting %∗ on L∞(Ω,μ) such that %(1A) = 1%(A) for A ∈ Σ and vice versa (cf. [14, pp. 35, 36]). For simplicity we write %∗ = % throughout. A Radon measure μ on X is a nonnegative real-valued Borel measure on B(X) such that for each Borel set E in B(X), μ(E) = sup{μ(K) : K ⊆ E, K compact} . A Borel measure μ on X is called: (i) a category measure iff the Borel null sets and the Borel sets of first category are the same. Then (X,B∧(X), μ) is called a category measure space (cf. Oxtoby [28], p. 86); (ii) regular iff it satisfies one of the following equivalent conditions: (I) μ(B) = sup{μ(F ) : F ⊆ B, F closed}, (II) μ(B) = inf{μ(U) : B ⊆ U, U open},