We extend some topologies on the space of upper semicontinuous functions with compact support to those on that of general upper semicontinuous functions and see that graphical topology and modified L topology are the same. We then define random upper semicontinuous functions using their topological Borel field and finally give a Choquet theorem for random upper semicontinuous functions.

gC(μ) := { F0(C) Fμ(C) if Fμ(C) > 0 ; +∞ if Fμ(C) = 0. Then fC is an upper semicontinuous function: if μ ∈ SM is such that Fμ(C) > 0, then fC is continuous at point μ. Otherwise, fC(μ) = ∞ and fC is trivially upper semicontinuous at point μ. Clearly, one has G(μ) = infC∈C fC(μ) ; as an infimum of upper semicontinuous functions, it is itself upper semicontinuous, and therefore attains its maximu...

Using the method of generic continuity of set-valued mappings, this paper studies the stability of weakly Pareto-Nash and Pareto-Nash equilibria for multiobjective population games, when payoff functions are perturbed. More precisely, the paper investigates the continuity properties of the set of weakly Pareto-Nash equilibria and that of the set of Pareto-Nash equilibria under sufficiently smal...

Below let II = [0, 1]. A well-known topological theorem due to Katětov states: Suppose (X, τ) is a normal topological space, and let f : X → II be upper semicontinuous, g : X → II be lower semicontinuous, and f ≤ g. Then there is a continuous h : X → II such that f ≤ h ≤ g. Recall that f : X → II is upper semicontinuous if f is continuous from (X, τ) to (II, ω); lower semicontinuous if continuo...

Katětov-Tong insertion type theorem For every upper semicontinuous real function f and lower semicontinuous real function g satisfying f ≤ g, there exists a continuous real function h such that f ≤ h ≤ g (Katětov 1951, Tong 1952). For every lower semicontinuous real function f and upper semicontinuous real function g satisfying f ≤ g, there exists a continuous real function h such that f ≤ h ≤ ...

Flesch et al [3] showed that, if the payoff functions are bounded and lower semicontinuous, then such a game always has a pure, subgame perfect -equilibrium for > 0. Here we prove the same result for bounded, upper semicontinuous payoffs. Moreover, Example 3 in Solan and Vieille [7] shows that if one player has a lower semicontinuous payoff and another player has an upper semicontinuous payoff,...

In the paper we study the continuity properties of the solution set of upper semicontinuous differential inclusions in both regularly and singularly perturbed case. Using a kind of dissipative type of conditions introduced in [1] we obtain lower semicontinuous dependence of the solution sets. Moreover new existence result for lower semicontinuous differential inclusions is proved.