Abstract. A quasi-state is a positive functional on C(X) that is only assumed to be linear on singly-generated subalgebras. We consider the "iterated integral" of two quasi-states and determine when this gives a quasi-state on the product space. We also provide explicit formulas for the corresponding quasi-measures in case it does. Finally, we show the general failure of Fubini's Theorem for quasi-states.If X is a compact, Hausdorff space, we let C(X) denote the collection of realvalued continuous functions on X. We let sp f denote the range of f . A quasi-state is a function ρ : C(X) → R such that:(ii) ρ(1) = 1.(iii) If r ∈ R, then ρ(rf ) = rρ(f ).

Abstract. Let X be a compact Hausdorff space and let A denote the subsets of X which are either open or closed. A quasi-linear functional is a map ρ : C(X) → R which is linear on singly generated subalgebras and such that |ρ(f )| ≤ M f for some M < ∞. There is a one-to-one correspondence between the quasi-linear functional on C(X) and the set functions µ :The space of quasi-linear functionals is investigated and quasi-linear maps between two C(X) spaces are studied.Let X be a compact Hausdorff space and C(X) the space of real-valued continuous functions on X. A map ρ : C(X) → R is said to be a quasi-linear functional if ρ is linear on singly generated subalgebras and bounded in the sense that there exists an M < ∞ such that |ρ(f )| ≤ M f u for all f ∈ C(X). Let ρ be the minimal such M . If ρ and η are quasi-linear functionals, we define ρ + η by pointwise action on functions. In this fashion, the collection of all quasi-linear functionals becomes a normed linear space. Call this space QL(X).Notice that if ρ is quasi-linear, and fg = 0, then ρ(f + g) = ρ(f) + ρ(g). In fact, if f and g are also positive, we have that the subalgebra generated by f − g contains both f and g. In general, we can break f and g into positive and negative parts to get the result. Also notice that if c is a constant,Our goal is to find set functions that produce all quasi-linear functionals on C(X). We will use an approach inspired by the techniques in [1] where the theory of positive quasi-linear functionals is presented. We use the notation f ≺ U when U is open to state that 0 ≤ f ≤ 1 and f has support contained in U . We also use the notation sp f for the image of f .Let O be the collection of open sets in X and C the collection of closed sets. Also, let A = O ∪ C. Thus A is the collection of subsets of X which are either open or closed.

since the kernel of T , (sin x)/x, has a derivative which does not decay quickly enough at infinity to apply the usual theory (see Davis and Chang [3]). Our aim in this paper is to show a result on singular integrals which in fact does include operators defined with kernels such as (sin x)/x.Our result will be a variant of a classical result of Calderón and Zygmund [1]. Actually, the statement will resemble that of a theorem found in Stein's [5] treatment of the Calderón and Zygmund theory. In order to state our results succinctly, we first introduce a little terminology.We say that a function p ≥ 0 on R q satisfies a reverse-L ∞ inequality (abbreviated as "p satisfies RL ∞ ") if there is a constant C such that for every cube Q ⊆ R q centered at the origin we have 0 < p| Q ∞ ≤ Cp Q . Here and throughout, p| Q denotes the restriction of the function p to Q and p Q denotes the average of the function p on Q. With these notations and conventions, we can now state our results.

Abstract. We give a direct proof that the space of Baire quasi-measures on a completely regular space (or the space of Borel quasi-measures on a normal space) is compact Hausdorff. We show that it is possible for the space of Borel quasi-measures on a non-normal space to be non-compact. This result also provides an example of a Baire quasi-measure that has no extension to a Borel quasi-measure. Finally, we give a concise proof of the Wheeler-Shakmatov theorem, which states that if X is normal and dim(X) ≤ 1, then every quasi-measure on X extends to a measure.

Abstract. A quasi-measure is a non-subadditive measure defined on only open or closed subsets of a compact Hausdorf space. We investigate the nature of irreducible partitions as defined by Aarnes and use the results to construct quasi-measures when g(X) = 1. The cohomology ring is an important tool for this investigation.