In 1965, Njastad [2] introduced the notion of α-sets in topological space. In 1983, Mashhour et al. [1] introduced, with the help of α-sets, a weak form of continuity which they termed as α-continuity. Noiri [3] introduced the same concept, but under the name strong semicontinuity. Noiri [4] defined with the aid of α-sets a new weakened form of continuous mapping called weakly α-continuous mapp...

Given a weakly convergent sequence of positive functions in W 1,p 0 (Ω), we prove the equivalence between its convergence in the sense of obstacles and the lower semicontinuity of the term by term duality product associated to (the p-Laplacian of) weakly convergent sequences of p-superharmonic functions of W 1,p 0 (Ω). This result implicitly gives new characterizations for both the convergence ...

We show that the value of a zero-sum Bayesian game is a Lipschitz continuous function of the players’common prior belief, with respect to the total variation metric on beliefs. This is unlike the case of general Bayesian games, where lower semi-continuity of Bayesian equilibrium (BE) payo¤s rests on the "almost uniform" convergence of conditional beliefs. We also show upper semi-continuity (USC...

In conclusion, I may point out that Theorem II may be generalized by a weakening of the hypothesis (d) . (1) In the first place, continuity of [a, g] f with respect to the pair of variables a, /3 may be replaced by upper semi-continuity . This generalization requires no change in the proof . (2) This continuity (or upper semi-continuity) with respect to (a, /3) is used only to show that the set...

X ω = 1. An upper semicontinuous function φ : X → [−∞,+∞) is called ω-plurisubharmonic (ω-psh) if φ ∈ L(X) and ωφ := ω + dd φ ≥ 0. By PSH(X,ω) (resp. PSH(X,ω)) we denote the set of ω-psh (resp. negative ω-psh) functions on X . The complex Monge-Ampère equation ω u = fω n was solved for smooth positive f in the fundamental work of S. T. Yau (see [Yau]). Later S. Kolodziej showed that there exist...

Let (Ek)k∈N be a sequence of differential information economies, converging to a limit differential information economy E∞ (written as Ek ⇒ E∞). Denote by C (Ek) the set of all -private core allocations, ≥ 0 (for = 0 we get the private core of Yannelis (1991), denoted by C(Ek)). Under appropriate conditions, we prove the following stability results: (1) (upper semicontinuity): if Ek ⇒ E∞, fk ∈ ...

We establish uniform continuity of the value for zero-sum games with differential information, when the distance between changing information fields of each player is measured by the Boylan pseudometric. We also show that the optimal strategy correspondence is upper semicontinuous when the information fields of players change (even with the weak topology on players’ strategy sets), and is appro...

For an upper semi-continuous set-valued mapping from one topological space to another and for a lower semi-continuous function defined on the product of these spaces, Berge’s theorem states lower semi-continuity of the minimum of this function taken over the image sets. It assumes that the image sets are compact. For Hausdorff topological spaces, this paper extends Berge’s theorem to set-valued...