C. Ionescu Tulcea

On the Approximation of Upper Semi-Continuous Correspondences and the Equilibriums of Generalized Games

In Section 1 we explain some of the definitions and terminology that we use. In Section 2 we prove several theorems concerning the approximation of upper semi-continuous correspondences having for range a locally convex space. Theorems 1 and 2 (and Corollary 1) generalize certain approximation theorems by G. Haddad [16, pp. 1352213541, G. Haddad and J. M. Lasry [ 17, pp. 299-3001, and J. P. Aubin and A. Cellina [2, pp. 86891 (see also F. S. De Blasi [lo] and J. M. Lasry and R. Robert [21]). Although some of the details of the proofs of Theorems 1 and 2 are new, the basic ideas are taken from the above mentioned papers. Theorem 3 shows that the correspondences in the approximating families can be chosen so that they are regular (see Section 1). This theorem (see the remark following its proof) contains a classical result of M. Hukuhara [ 18, pp. 56573. Theorem 3 is proved using Theorem 2 and Propositions 1 and 4. Variants of Propositions 2, 3, and 4 were given in [ 193. In Section 3 we give, among others, Theorem 5, which concerns the existence of equilibriums of generalized games ( = abstract economies). The main purpose of this result is to replace the continuity hypothesis in the W. Shafer-H. Sonnenschein equilibrium theorem for generalized games by an upper semi-continuity one. The proof of Theorem 5 is based on the results on the approximation of upper semi-continuous correspondences obtained in Section 2. Theorem 5 us used in Section 4. In Section 4 we establish Theorems 6 and 7. These theorems show that certain statements concerning the equilibrium of generalized games are equivalent to certain statements concerning minimax inequalities of K. Fan type. Theorems 1, 2, 3, 5, 6, and 7 are the main results of this paper. 267 0022-247X/88

Ergodic theorems, I

3.00

Liftings for abstract valued functions and separable stochastic processes

1 Strong laws of large numbers . . . . . . . . . . . . . . . . . . 1 2 Heuristics and proof for the Ergodic theorem . . . . . . . . . 3 3 Stationary processes (Theorem <2>) . . . . . . . . . . . . . . 5 4 The strong law of large numbers (Theorem <1>) . . . . . . . 6 5 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 6 The subadditive ergodic theorem . . . . . . . . . . . . . . . . 9 7 An application . . . . . . . . . . . . . . . . . . . . . . . . . . 12

On liftings and derivation bases

In this paper we extend the notion of lifting from real-valued functions to abstract-valued functions (functions with values in a completely regular space). We prove the existence and uniqueness of the “abstract lifting” associated with a given “real lifting”. As an application we show that given a stochastic process with values in a completely regular space, the modification of this process obtained by applying a lifting is always separable.

Extensions of certain operators to spaces of abstract integrable functions

It was shown by DieudonnC in [5], that the existence of a derivation basis, with certain properties, implies the existence of a lifting. At the time of the publication of [5] it was not known however, whether or not liftings always exist, hence it was not known whether or not derivation bases with convenient properties always exist. We know now that in the case of strictly localizable spaces, liftings do exist (see [16], Chap. IV). Recently D. Kiilzow has shown (see [17]), using liftings, that derivation bases with convenient properties also exist, in the case of strictly localizable spaces (see also [l 81). In this paper we shall establish further relations between liftings and derivation bases. In the first section we gather various definitions and we give a criterion for a derivation basis to be strong. In the second section we discuss derivation bases associated with a lower density. In particular we show that certain differentiability results can be obtained directly using the topologies associated with lower densities and with liftings (see [I l] and [16], Chap. V). Section three deals with derivation bases on locally compact spaces and section four with strong Iiftings and derivation bases satisfying condition (C). In section five we show that for every locally compact group there exist approximate identities yielding pointwise convergence. For this we use liftings commuting with the left translations of the considered group (see [15]). Results of this type were obtained by Edwards and Hewitt (see [6]) for certain locally compact groups and were used to obtain various pointwise inversion formulas for the Fourier transform (see also the remarks at the end of section 5). In section six we give a result concerning lower densities and derivation bases commuting with certain groups of mappings.

The existence of a lifting

In this paper we discuss the extension of operators onL1R spaces to operators onL1E andP1E spaces (see Section 1), whereE is a Banach space. A necessary and sufficient condition for the existence of the extension to a spaceP1E is given (see Section 3) whenE has the weak Radon-Nikodym property. The paper contains certain applications to ergodic theory and a theorem giving a characterization of weakly conditionally compact sets.

Domination of measures and disintegration of measures

Throughout this chapter we assume that N = \(\bar N\). With the exception of sections 3 and 4 we suppose that (X,N,ℛ) is strictly localizable and that l is a fixed partition of X consisting of non-negligible integrable sets satisfying sup{\(tilde K\)|Kєl} = \(tilde X\).

On certain endomorphisms of L_R^\infty (Z,\mu )

Throughout this chapter the setting is that of compact or locally compact spaces and positive Radon measures. We make constant use of the notation and terminology of chapter 8.

Integrability and measurability for abstract valued functions

In this chapter, unless we mention explicitly the contrary, we shall denote by ℒ1 = (Z1,µ1) and ℒ2 = (Z2,µ2) two objects, where Z1 and Z2 are locally compact spaces, µ1 ≠ 0 is a positive Radon measure on Z1 and µ1 ≠ 0 a positive Radon measure on Z2.

Measure and integration

In this chapter we shall discuss the integrability and measurability of functions with values in a Banach space. The definitions and results of sections 1, 2 and 3 are somewhat similar to those in chapter 1. The definitions and results of sections 4, 5, 6 are based on the notion of lifting and are essential for the applications in the next chapter.