[Paper Review] Convex functions on dual Orlicz spaces

Freddy Delbaen, Keita Owari|arXiv (Cornell University)|Nov 18, 2016

Advanced Banach Space Theory10 references4 citations

TL;DR

This paper establishes that convex functions on the dual Orlicz space $L^*_{\Phi}$ are lower semicontinuous (lsc) for the Mackey topology $\tau(L^*_{\Phi}, L_{\Phi})$ if and only if they are lsc with respect to convergence in probability on every order interval $[-\zeta, \zeta]$. The key contribution is a Komlós-type theorem for $L^*_{\Phi}$, showing that every norm-bounded sequence in $L^*_{\Phi}$ admits a sequence of forward convex combinations converging almost surely and uniformly bounded in $L^*_{\Phi}$, which enables the characterization of Mackey topology lsc and continuity via probabilistic convergence.

ABSTRACT

In the dual $L_{Φ^*}$ of a $Δ_2$-Orlicz space $L_Φ$, that we call a dual Orlicz space, we show that a proper (resp. finite) convex function is lower semicontinuous (resp. continuous) for the Mackey topology $τ(L_{Φ^*},L_Φ)$ if and only if on each order interval $[-ζ,ζ]=\{ξ: -ζ\leq ξ\leqζ\}$ ($ζ\in L_{Φ^*}$), it is lower semicontinuous (resp. continuous) for the topology of convergence in probability. For this purpose, we provide the following Komlós type result: every norm bounded sequence $(ξ_n)_n$ in $L_{Φ^*}$ admits a sequence of forward convex combinations $\barξ_n\in\mathrm{conv}(ξ_n,ξ_{n+1},...)$ such that $\sup_n|\barξ_n|\in L_{Φ^*}$ and $\barξ_n$ converges a.s.

Motivation & Objective

• To characterize the lower semicontinuity and continuity of convex functions on the dual Orlicz space $L^*_{\Phi}$ with respect to the Mackey topology $\tau(L^*_{\Phi}, L_{\Phi})$.
• To establish a Komlós-type theorem in the dual of a $\Delta^2$-Orlicz space, ensuring almost sure convergence of forward convex combinations of norm-bounded sequences.
• To show that for proper convex functions on $L^*_{\Phi}$, $\tau(L^*_{\Phi}, L_{\Phi})$-lsc is equivalent to lsc on order intervals under convergence in probability.
• To apply the results to monetary utility functions and convex risk measures, characterizing their regularity under the Mackey topology.

Proposed method

• Prove a Komlós-type result: every norm-bounded sequence in $L^*_{\Phi}$ admits a sequence of forward convex combinations $\bar{\xi}_n \in \text{conv}(\xi_n, \xi_{n+1}, \dots)$ such that $\sup_n |\bar{\xi}_n| \in L^*_{\Phi}$ and $\bar{\xi}_n \to \xi$ a.s.
• Use the Kreïn-Šmulian theorem to link sequential properties in the Mackey topology to topological closedness in order intervals.
• Characterize $\tau(L^*_{\Phi}, L_{\Phi})$-lower semicontinuity via sequential convergence in probability on order intervals $[-\zeta, \zeta]$.
• Establish equivalence between $\tau(L^*_{\Phi}, L_{\Phi})$-continuity and continuity under convergence in probability on bounded sets, using the dual representation of convex functions.
• Apply the results to monetary utility functions, showing that upper semicontinuity is equivalent to continuity from above, and continuity to continuity from below.
• Use the Dunford-Pettis theorem and Moreau's theorem to prove continuity results in the dual space setting.

Experimental results

Research questions

• RQ1 When is a proper convex function on $L^*_{\Phi}$ lower semicontinuous for the Mackey topology $\tau(L^*_{\Phi}, L_{\Phi})$?
• RQ2 Can the Mackey topology lsc property be characterized by convergence in probability on order intervals in $L^*_{\Phi}$?
• RQ3 Does every norm-bounded sequence in $L^*_{\Phi}$ admit a sequence of forward convex combinations that converges almost surely and remains bounded in $L^*_{\Phi}$?
• RQ4 How do these results characterize the regularity of monetary utility functions and convex risk measures under the Mackey topology?
• RQ5 Is the weak* closure of an order-closed convex set in $L^*_{\Phi}$ equal to its sequential weak* closure?

Key findings

• A proper convex function $f$ on $L^*_{\Phi}$ is $\tau(L^*_{\Phi}, L_{\Phi})$-lower semicontinuous if and only if it is lower semicontinuous on every order interval $[-\zeta, \zeta]$ with respect to convergence in probability.
• For any norm-bounded sequence $ (\xi_n) $ in $ L^*_{\Phi} $, there exists a sequence of forward convex combinations $ \bar{\xi}_n \in \text{conv}(\xi_n, \xi_{n+1}, \dots) $ such that $ \sup_n |\bar{\xi}_n| \in L^*_{\Phi} $ and $ \bar{\xi}_n \to \xi $ almost surely.
• A convex set $ C \subset L^*_{\Phi} $ is $ \sigma(L^*_{\Phi}, L_{\Phi}) $-closed if and only if $ C \cap [-\zeta, \zeta] $ is closed in $ L^0 $ for every $ \zeta \in L^*_{\Phi} $.
• A finite convex function $ f $ on $ L^*_{\Phi} $ is $ \tau(L^*_{\Phi}, L_{\Phi}) $-continuous if and only if it is continuous with respect to convergence in probability on bounded sets.
• For monetary utility functions $ u $, $ \tau(L^*_{\Phi}, L_{\Phi}) $-upper semicontinuity is equivalent to continuity from above: $ \xi_n \downarrow \xi \Rightarrow u(\xi) = \lim_n u(\xi_n) $.
• For finite monetary utility functions, $ \tau(L^*_{\Phi}, L_{\Phi}) $-continuity is equivalent to continuity from below: $ \xi_n \uparrow \xi \Rightarrow u(\xi) = \lim_n u(\xi_n) $.