Commutative L-algebras and measure theory

Measure and integration theory for finitely additive measures, including vector-valued measures, is shown to be essentially covered by a class of commutative L-algebras, called measurable algebras. The domain and range of any measure is a commutative L-algebra. Each measurable algebra embeds into its structure group, an abelian group with a compatible lattice order, and each (general) measure extends uniquely to a monotone group homomorphism between the structure groups. On the other hand, any measurable algebra X is shown to be the range of an essentially unique measure on a measurable space, which plays the role of a universal covering. Accordingly, we exhibit a fundamental group of X, with stably closed subgroups corresponding to a special class of measures with X as target. All structure groups of measurable algebras arising in a classical context are archimedean. Therefore, they admit a natural embedding into a group of extended real-valued continuous functions on an extremally disconnected compact space, the Stone space of the measurable algebra. Extending Loomis’ integration theory for finitely additive measures, it is proved that, modulo null functions, each integrable function can be represented by a unique continuous function on the Stone space.

Legitimate equilibrium

We present a general existence result for a type of equilibrium in normal-form games, which extends the concept of Nash equilibrium. We consider nonzero-sum normal-form games with an arbitrary number of players and arbitrary action spaces. We impose merely one condition: the payoff function of each player is bounded. We allow players to use finitely additive probability measures as mixed strategies. Since we do not assume any measurability conditions, for a given strategy profile the expected payoff is generally not uniquely defined, and integration theory only provides an upper bound, the upper integral, and a lower bound, the lower integral. A strategy profile is called a legitimate equilibrium if each player evaluates this profile by the upper integral, and each player evaluates all his possible deviations by the lower integral. We show that a legitimate equilibrium always exists. Our equilibrium concept and existence result are motivated by Vasquez (2017), who defines a conceptually related equilibrium notion, and shows its existence under the conditions of finitely many players, separable metric action spaces and bounded Borel measurable payoff functions. Our proof borrows several ideas from (Vasquez (2017)), but is more direct as it does not make use of countably additive representations of finitely additive measures by (Yosida and Hewitt (1952)).

On Four Classical Measure Theorems

A subset B of an algebra A of subsets of a set Ω has property (N) if each B-pointwise bounded sequence of the Banach space ba(A) is bounded in ba(A), where ba(A) is the Banach space of real or complex bounded finitely additive measures defined on A endowed with the variation norm. B has property (G) [(VHS)] if for each bounded sequence [if for each sequence] in ba(A) the B-pointwise convergence implies its weak convergence. B has property (sN) [(sG) or (sVHS)] if every increasing covering {Bn:n∈N} of B contains a set Bp with property (N) [(G) or (VHS)], and B has property (wN) [(wG) or (wVHS)] if every increasing web {Bn1n2⋯nm:ni∈N,1≤i≤m,m∈N} of B contains a strand {Bp1p2⋯pm:m∈N} formed by elements Bp1p2⋯pm with property (N) [(G) or (VHS)] for every m∈N. The classical theorems of Nikodým–Grothendieck, Valdivia, Grothendieck and Vitali–Hahn–Saks say, respectively, that every σ-algebra has properties (N), (sN), (G) and (VHS). Valdivia’s theorem was obtained through theorems of barrelled spaces. Recently, it has been proved that every σ-algebra has property (wN) and several applications of this strong Nikodým type property have been provided. In this survey paper we obtain a proof of the property (wN) of a σ-algebra independent of the theory of locally convex barrelled spaces which depends on elementary basic results of Measure theory and Banach space theory. Moreover we prove that a subset B of an algebra A has property (wWHS) if and only if B has property (wN) and A has property (G).

The Time Concept in Atonic and Sub-Atonic Systens - Reconciliation of the Time-Reversal-Invariance and the Macroscopic Arrow of Time

A new conception of time is presented in the framework of the quantum generalized stochastic and infinitely divisible fields. A non-unitary evolution operator lacking the continuous group property is derived from a time-reversal-invariant field theory in Minkowski's space. It describes the arrow of time on the quantum level. By quantizing the field action integral the usual evolution operator is obtained as a particular case. Quantum processes violating the T-symmetry are possible in the present theory. It is also explained why Born's interpretation of the wave function is necessary. The Feynman path integral is obtained as the limit of a series of similar integrals with finitely additive measures. This form of the Feynman integral does not conflict on the quantum level Heisenberg's Uncertainty Principle.