Countably Additive Research Articles

Visualizing a Nonmeasurable Set

In most real analysis textbooks, the standard example of a nonmeasurable set is a subset of the real line that is due to Vitali [3]. We describe a similar nonmeasurable subset of the torus (and hence the plane), where we can more easily visualize the set. In the process of constructing the set, students get an opportunity to experience how topics they learned in algebra and topology can be used in analysis. The idea of Vitali's example is to express the unit interval I as a disjoint union of countably many mutually congruent sets Ak. The nonmeasurability of each Ak follows from the observation that I = UkEz Ak and that countable additivity of measure implies that 1 = m(I) = Lkez m(Ak). Since each set Ak must have the same measure, the last equation shows that no nonnegative value can be assigned as the measure of each Ak. We will use this same idea with the square [0, 1] x [0, 1] in the plane R2. The advantage is that we will now have a more visual object than that of Vitali's example because the example will appear as a subset of a torus.

Valuation in infinite-horizon sequential markets with portfolio constraints

We develop a theory of valuation of assets in sequential markets over an infinite horizon and discuss implications of this theory for equilibrium under various portfolio constraints. We characterize a class of constraints under which sublinear valuation and a modified present value rule hold on the set of non-negative payoff streams in the absence of feasible arbitrage. We provide an example in which valuation is non-linear and the standard present value rule fails in incomplete markets. We show that linearity and countable additivity of valuation hold when markets are complete. We present a transversality constraint under which valuation is linear and countably additive on the set of all payoff streams regardless of whether markets are complete or incomplete.

Vitali's Theorem and WWKL

Continuing the investigations of X. Yu and others, we study the role of set existence axioms in classical Lebesgue measure theory. We show that pairwise disjoint countable additivity for open sets of reals is provable in RCA0. We show that several well-known measure-theoretic propositions including the Vitali Covering Theorem are equivalent to WWKL over RCA0.

On the modulus of measures with values in topological Riesz spaces

The paper is devoted to a study of some aspects of the theory of (topological) Riesz space valued measures. The main topics considered are the following. First, the problem of existence (and, particularly, the so-called proper existence) of the modulus of an order bounded measure, and its relation to a similar problem for the induced integral operator. Second, the question of how properties of such a measure like countable additivity, exhaustivity or socalled absolute exhaustivity, or the properties of the range space, influence the properties of the modulus of the measure. Third, the problem of exhibiting (or constructing) Banach lattices that are “good” in many respects, and yet admit a countably additive measure whose modulus is not countably additive. A few applications to weakly compact operators from spaces of bounded measurable functions to Banach lattices are also presented.

Drift conditions and invariant measures for Markov chains

We consider the classical Foster–Lyapunov condition for the existence of an invariant measure for a Markov chain when there are no continuity or irreducibility assumptions. Provided a weak uniform countable additivity condition is satisfied, we show that there are a finite number of orthogonal invariant measures under the usual drift criterion, and give conditions under which the invariant measure is unique. The structure of these invariant measures is also identified. These conditions are of particular value for a large class of non-linear time series models.

OUTER MEASURES GENERATED BY A COUNTABLY ADDITIVE MEASURE ON A RING OF SETS

Let $\mathcal{R}$ be any ring of subsets of a set $X$ which is not an algebra and let $\mathcal{A}$ be the algebra generated by $\mathcal{R}$. Suppose that $\mu$ is a countably additive measure on $\mathcal{R}$ and that $\mu^*$ is the outer measure generated by $(\mu,\mathcal{R})$. If $X$ is a countable union of sets in $\mathcal{R}$, then there is a unique countably additive measure $\nu$ on $\mathcal{A}$ which extends $\mu$, and the outer measure generated by $(\nu,\mathcal{A})$ coincides with $\mu^*$. If $X$ is not a countable union of sets in $\mathcal{R}$, then there exists a family $\{ \mu_p : 0 \leq p \leq \infty \}$ of countably additive measures on $\mathcal{A}$ such that each $\mu_p$ agrees with $\mu$ on $\mathcal{R}$. For $0 \leq p \leq \infty$, let $\mu_p^*$ denote the outer measure generated by $(\mu_p, \mathcal{A})$. Then we have $\mu_0^* \leq \mu_p^* \leq \mu_q^* \leq \mu_\infty^* =\mu^*$ for $0< p < q < \infty$. Moreover, if $\mathcal{M}$ and $\mathcal{M}_p$, respectively, denotes the $\sigma$-algebra of $\mu^*$-measurable and $\mu_p^*$-measurable sets, then $\mathcal{M}_p = \mathcal{M}_1 \subset \mathcal{M}_0 = \mathcal{M}_\infty = \mathcal{M}$ for all positive real numbers $p$. As examples, we give countably additive measures on rings for which $\mathcal{M} = \mathcal{M}_1$ and $\mathcal{M} \neq \mathcal{M}_1$, respectively. By the outer measures generated by $\mu$ we shall mean the outer measures $\mu^*$ and $\mu_p^*$ $(0 \leq p \leq \infty)$.

Asset price bubbles in Arrow-Debreu and sequential equilibrium

Price bubbles in an Arrow-Debreu equilibrium in an infinite-time economy are a manifestation of lack of countable additivity of valuation of assets. In contrast, the known examples of price bubbles in a sequential equilibrium in infinite time cannot be attributed to the lack of countable additivity of valuation. In this paper we develop a theory of valuation of assets in sequential markets (with no uncertainty) and study the nature of price bubbles in light of this theory. We define a payoff pricing operator that maps a sequence of payoffs to the minimum cost of an asset holding strategy that generates it. We show that the payoff pricing functional is linear and countably additive on the set of positive payoffs if and only if there is no Ponzi scheme, provided that there is no restriction on long positions in the assets. In the known examples of equilibrium price bubbles in sequential markets valuation is linear and countably additive. The presence of a price bubble means that the dividends of an asset can be purchased in sequential markets at a cost lower than the asset's price. We present further examples of equilibrium price bubbles in which valuation is nonlinear, or linear but not countably additive.

ON THE EXTENSION OF SET-VALUED SET FUNCTIONS

We extend additive set-valued set functions and normal multimeasures defined on a ring of subsets. We also prove a Carathéodory-Hahn-Kluvanek-type theorem for additive set-valued set functions. Finally, we establish results on the extension of transition multimeasures.

Purely Game-theoretic Random Sequences: I. Strong Law of Large Numbers and Law of the Iterated Logarithm

Random sequences are usually defined with respect to a probability distribution~$\bf P$ (a $\sigma$-additive set function, normed to one, defined over a $\sigma$-algebra) assuming Kolmogorov's axioms for probability theory. In this paper, without using this axiomatics, we give a definition of random (typical) sequences taking as primitive the notion of a martingale and using the principle of the excluded gambling strategy. In this purely game-theoretic framework, no probability distribution or, partially or fully specified, system of conditional probability distributions needs to be introduced. For these typical sequences, we prove direct algorithmic versions of Kolmogorov's strong law of large numbers (SLLN) and of the upper half of Kolmogorov's law of the iterated logarithm~(LIL).

First Alexandroff Decomposition Theorem for Topological Lattice Group Valued Measures

Let (X, F) be an Alexandroff space, let A(F) be the Boolean subalgebra of 2X generated by F, let G be a Hausdorff commutative topological lattice group and let rbaF(A(F), G) denote the set of all order bounded F-inner regular finitely additive set functions from A(F) into G. Using some special properties of the elements of rbaF(A(F), G), we extend to this setting the first decomposition theorem of Alexandroff.

Countable Additivity and Subjective Probability

While there are several arguments on either side, it is far from clear as to whether or not countable additivity is an acceptable axiom of subjective probability. I focus here on de Finetti's central argument against countable additivity and provide a new Dutch book proof of the principle, To argue that if we accept the Dutch book foundations of subjective probability, countable additivity is an unavoidable constraint.

Arbitrage, martingales and bubbles

Viability of security prices implies linear valuation of payoffs but, if there exist an infinite number of securities or trading dates, does not imply the existence of a risk-neutral probability since countable additivity may fail. An example is given.

Decoherence Functionals for von Neumann Quantum Histories: Boundedness and Countable Additivity

Gell–Mann and Hartle have proposed a significant generalisation of quantum theory in which decoherence functionals perform a key role. Verifying a conjecture of Isham–Linden–Schreckenberg, the author analysed the structure of bounded, finitely additive, decoherence functionals for a general von Neumann algebra A (where A has no Type I2 direct summand). Isham et al. had already given a penetrating analysis for the situation where A is finite dimensional. The assumption of countable additivity for a decoherence functional may seem more plausible, physically, than that of boundedness. The results of this note are obtained much more generally but, when specialised to L(H), the algebra of all bounded linear operators on a separable Hilbert space H, give:

Countably Additive Subjective Probabilities

The subjective probabilities implied by Savage's (1954, 1972) Postulates are finitely but not countably additive. The failure of countable additivity leads to two known classes of dominance paradoxes, money pumps and indifference between an act and one that pointwise dominates it. There is a common resolution to these classes of paradoxes and to any others that might arise from failures of countably additivity. It consists of reinterpreting finitely additive probabilities as the “traces” of countably additive probabilities on larger state spaces. The new and larger state spaces preserve the essential decision-theoretic structures of the original spaces.

Representation of preferences on fuzzy measures by a fuzzy integral

In recent research on decisions under uncertainty, the basic universe of choice has been extended from probability distributions (additive set functions) to set functions that are not necessarily additive. This family has been investigated in other contexts under the name of fuzzy measures, and there exists a theory of (fuzzy) integration for fuzzy measures. In the present paper this concept of an integral is used in a utility representation of preferences on the set of non-additive set functions. A system of axioms for such a representation is presented, and its usefulness with respect to decision theory is evaluated.

QUALITATIVE MODALITIES

We add a binary operator ≥ to the logical language, with intended meaning of φ&lt;ψ: ‘φ is at least as likely, probable, or trustworthy, as ψ’. The operator ≥ is interpreted on Kripke structures, making it possible to define the standard necessity operator □ in terms of ≥. The operator ≥ provides us with an intermediate for the K-axiom, in the sense that we have both □(p→q)→(q≥p) and (q≥p)→(□p→□q). We discuss two semantics for this binary modal operator. It turns out that, as shown by Gärdenfors and Segerberg, ≥ is not only too weak to distinguish finite models from infinite ones or to distinguish countable additivity from finite additivity, ≥ also cannot distinguish sophisticated ways of assigning exact probabilities to events (‘measuring’) from the conceptually simpler task of just counting them.

On the Doléans function of set-indexed submartingales

A necessary and sufficient condition for the countable additivity of the Doléans function of a set-indexed weak submartingale is given in terms of stopping sets.

Canonical Representation of Set Functions

The representation of a cooperative transferable utility game as a linear combination of unanimity games may be viewed as an isomorphism between no-necessarily additive set functions on the players space and additive ones on the coalitions space. (Or, alternatively, between nonadditive probability measures on a state space and additive ones on the space of events.) We extend the unanimity-basis representation to general (infinite) spaces of players, study spaces of games which satisfy certain properties and provide some conditions for α-additivity of the resulting additive set function (on the space at coalitions). These results also allow us to extend some representations of the Choquet integral from finite to infinite spaces.

On Shioda's problem about Jacobi sums II

In the present paper, we will give a complete affirmative answer to the l-part of Shioda’s problem ([5, Question 3.4]) on Jacobi sums J (a) l (p), and to the conjecture (F. Gouvea and N. Yui [1, Conjecture (1.9)]) which comes from Shioda’s problem and my congruences for Jacobi sums (see [3, Theorem 2]) (see Theorem 1 and its Corollary of the present paper). We retain the notation of [4], but l is any odd prime number here. Furthermore, let n be any positive integer and let ζm be a primitive mth root of unity in C for any positive integer m. Let Q be the algebraic closure of Q in C. We fix an algebraic closure Ql of Ql, and by a fixed imbedding Q ↪→ Ql we consider Q as a subfield of Ql. Let M be any finite unramified extension of Ql in Ql, and put Mn = M(ζln) and πn = ζln − 1. Then πn is a prime element of Mn. Let σ−1 ∈ G = Gal(Mn/M) (the Galois group of Mn over M) be such that ζ σ−1 ln = ζ −1 ln . Let ordMn denote the normalized additive valuation of Mn, and let Un = U(Mn) be the group of principal units in Mn: Un = U(Mn) = {x ∈Mn | ordMn(x− 1) ≥ 1}.

The countable additivity of set-valued integrals and F-valued integrals

Abstract In this paper, integrals of set-valued functions and F -valued functions on Banach space are further investigated, the countable additivity of the integrals is shown.