Strong singularity, disjointness, and strong finite additivity of finitely additive measures

AbstractThe most striking difference between finitely additive measures and countably additive measures is that the Hahn decomposition theorem may be only approximate if countable additivity fails. A corollary is that whereas two singular countably additive measures are disjoint in giving full measure to disjoint sets, this is usually only approximately true without countable additivity. One natural question is “Which finitely additive measures are disjoint from all singular measures?” In the Stone space setting, the question becomes “Which Radon measures have support disjoint from all singular Radon measures?” If so, the support is a P1-set of Atalla which is more general than a P-set. Construction of such measures is considered at length. Existence of nonmolecular nonatomic measures based on P1-points with support a P1-set is equivalent to the existence of a P1-homeomorph of βN in the Stone space. If (X, Σ, μ) is a positive localizable measure space, then BA(Σ) has L∞∗(X, Σ, μ) as a band and and L1(X, Σ, μ) as a subband. It is shown that the band complementary to L∞∗(X, Σ, μ) consists of measures disjoint to μ so supp(μ) is a P1-set in the Stone space of Σ. A notion of strong singularity to μ intermediate to disjointness and singularity is shown to be equivalent to strong finite additivity (admitting countable partitions by null sets) for elements of L∞∗(X, Σ, μ). The band of L∞∗(X, Σ, μ) orthogonal to L1(X, Σ, μ) consists of strongly finitely additive measures precisely when the measure algebra Σμ satisfies the λ-chain condition for a nonreal valued measurable cardinal λ or when the maximal ideal space of L∞(X, Σ, μ) does not have real valued measurable cellularity. A μ in BA(Σ) has a countably additive ideal of negligible sets iff it is not strongly finitely additive on any set of positive measure. On the Stone space, this corresponds to Radon measures with P-sets as supports. Even for purely finitely additive measures of this type, much of countable additive measure theory holds, for null functions are precisely those with null support. It is not known for Σ = 2x whether purely finitely additive measures of this type exist. Their existence may be independent of ZFC and/or other axioms such as Martin's, real-valued measurable cardinal, or measurable cardinal. A locally compact space Y is sham-compact so that C(X) = Cb(X) = Cc(X) precisely when its remainder in any compactification is a P-set. In this case M(X) = Mb(x) = Mc(X). The latter strong of identities is the definition of sham-sham compact spaces. These are characterized by having a P1-set as remainder in any compactification. New examples of sham-compact open subsets of the maximal ideal space Zμ of a positive localizable measure space (X, Σ, μ) are given based on Maharam's homogeneity character. However, no open subset of Zμ can be sham-sham compact without being sham compact. This is true when Zμ is replaced by any compact space which locally the support of a Radon measure