A Nonstandard Characterization of Weak Convergence

Abstract

Abstract. Let A" be any topological space, and C(X) the space of bounded continuous functions on X. We give a nonstandard characterization of weak convergence of a net of bounded linear functionals on CiX) to a tight Baire measure on X. This characterization applies whether or not the net or the individual functionals in the net are tight. Moreover, the characterization is expressed in terms of the values of an associated net of countably additive measures on all Baire sets of X; no distinguished family, such as the family of continuity sets of the limit, is involved. As a corollary, we obtain a new proof that a tight set of measures is relatively weakly compact.

Market clearing, utility functions, and securities prices

“…In this setting, the existence of equilibrium follows immediately from Robinson's Transfer Principle and results on existence of equilibrium in GEI models with a finite number of dates and states. 4 We show that the short-sale constraint does not bind at equilibrium. Consequently, as in Magill and Quinzii, we can use the first order conditions to characterize the equilibrium prices.…”

“…Since the binomial distribution converges in distribution to the normal distribution, Anderson and Rashid [4] …”

Infinitesimal methods in control theory: Deterministic and stochastic

“…Using the nonstandard criterion for compactness, it is enough to show that ~ ~'Y is nearstandard for every Vc*~V, Y~*K. Let ~cv, v be a solution to Equation *(7.3)v(a) with Y constant; i,e., d~ :v'Y = *gv(r, ~v.Y, y) dB~.Arguing exactly as in the proof of Lemma 7.11 we see that ~v,v is a.s. 55-continuous. Thus, ()t lV'Y)L(st-I(~a)) = 1, which shows (as in[4,17]) that h, v,Y isnearstandard with standard part (A v'Y)L(st-a(. )).…”

“…We make frequent use of Anderson's Lusin Recall the following characterisation of weak standard parts of measures: [4,18]. Let X be Hausdorff; an internal Baire probability measure v on * X is nearstandard (in the weak topology) iff vL(ns(* X)) = 1; in which case °v(A) = vL(st-l(A)) for Baire sets A in X.…”

“…Let V~ *72 and let v =°V as in the previous section. We have to show that /x v~/xv; by the Loeb-Anderson-Rashid characterisation of weak standard parts [4,18], it is sufficient to show that i x~': ixV(st-l(')). We will show that z v is a solution to Equations (7.1) for control v. If so, then…”