Convex functions on dual Orlicz spaces

Abstract

In the dual $L_{\varPhi ^*}$ of a $\varDelta _2$-Orlicz space $L_\varPhi$, 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 $\tau (L_{\varPhi ^*},L_\varPhi )$ if and only if on each order interval $[-\zeta ,\zeta ]=\{\xi : -\zeta \le \xi \le \zeta \}$ ($\zeta \in L_{\varPhi ^*}$), 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 $(\xi _n)_n$ in $L_{\varPhi ^*}$ admits a sequence of forward convex combinations ${{\bar{\xi }}}_n\in \text {conv}(\xi _n,\xi _{n+1},\ldots )$ such that $\sup _n|{\bar{\xi }}_n|\in L_{\varPhi ^*}$ and ${\bar{\xi }}_n$ converges a.s.

Appendix

Proof of Proposition 1.2

Only (3) $\Rightarrow$ (1) deserves a proof. (3) implies, by Proposition 1.1, $f=f^{**}$, and $|\mathbb {E}[\eta {\mathbb {1}}_A]|\le \frac{1}{n}\left( f(n{\mathbb {1}}_A)\vee f(-n{\mathbb {1}}_A)+c\right)$ for $A\in \mathcal {F}$ and $\eta \in L_1$ with $f^*(\eta )\le c$ by Young’s inequality; thus (3) implies that $\{\eta \in L_1: f^*(\eta )\le c\}$ is uniformly integrable, hence $\sigma (L_1,L_\infty )$-compact by the Dunford-Pettis theorem. Now Moreau’s theorem [17] shows that f is $\tau (L_\infty ,L_1)$-continuous. $\square$

Proof of Lemma 2.1

For each $\xi \in L_{\varPhi ^*}$, $\eta \mapsto \eta \xi$ continuously maps $(L_\varPhi ,\sigma (L_\varPhi ,L_{\varPhi ^*}))$ into $(L_1,\sigma (L_1,L_\infty ))$ since $\xi \zeta \in L_1$, $\forall \zeta \in L_\infty$. Thus if A is relatively $\sigma (L_\varPhi ,L_{\varPhi ^*})$-compact, its image $A\xi$ is relatively weakly compact in $L_1$, i.e. uniformly integrable. Conversely, if $A\xi$, $\xi \in L_{\varPhi ^*}$, are uniformly integrable, then $c_\xi :=\sup _{\eta \in A}\mathbb {E}[|\eta \xi |]<\infty$ for each $\xi \in L_{\varPhi ^*}$, so A is pointwise bounded in the algebraic dual $L_{\varPhi ^*}^\#$ of $L_{\varPhi ^*}$, and A is relatively $\sigma (L_1,L_\infty )$-compact in $L_1$. Thus if $(\eta _\alpha )_\alpha$ is a net in A with the pointwise limit $f(\xi )=\lim _\alpha \mathbb {E}[\eta _\alpha \xi ]$ in $L_{\varPhi ^*}^\#$, there is a unique $\eta _0\in L_1$ such that $f|_{L_\infty }(\xi )=\mathbb {E}[\eta _0\xi ]$ for $\xi \in L_\infty$. Then for each $\xi \in L_{\varPhi ^*}$, $\mathbb {E}[|\eta _0\xi |]=\sup _n\mathbb {E}[\eta _0\xi {\mathbb {1}}_{\{|\xi |\le n\}} \text {sgn}(\eta _0\xi )] =\sup _nf(\xi {\mathbb {1}}_{\{|\xi |\le n\}}\text {sgn}(\eta _0\xi )) \le c_\xi$, hence $\eta _0\in L_\varPhi$, while $|f(\xi )-f(\xi {\mathbb {1}}_{\{|\xi |\le n\}})|=|f(\xi {\mathbb {1}}_{\{|\xi |>n\}})|\le \sup _{\eta \in A}\mathbb {E}[|\eta \xi |{\mathbb {1}}_{\{|\xi |>n\}}] \rightarrow 0$ since $A\xi$ is uniformly integrable; hence $f(\xi )=\mathbb {E}[\eta _0\xi ]$. Therefore A is pointwise bounded and its $\sigma (L_{\varPhi ^*}^\#,L_{\varPhi ^*})$-closure in $L_{\varPhi ^*}^\#$ lies in $L_\varPhi$; hence A is relatively $\sigma (L_\varPhi ,L_{\varPhi ^*})$-compact.$\square$