I am interested in conditions for weak convergence on Orlicz spaces where the corresponding Young function, $\Phi:[0,\infty) \rightarrow [0,\infty)$, does not have the $\Delta_{2}$ condition: $\displaystyle \sup_{t \in [0,\infty)} \dfrac{\Phi(2t)}{\Phi(t)} < \infty.$

I have been attempting to apply the following theorem for Kosaku Yosida's book, "Functional Analysis" (see Chapter 5, Section 1 Theorem 3 on page 121):

Theorem 3: A sequence in a normed linear space $X$ converges weakly to an element $x_{\infty} \in X$ iff the following two conditions are satisfied: i.) $\sup_{n} \|x_{n}\| < \infty$, and ii.) $\lim_{n \rightarrow \infty} f(x_{n}) = f(x_{\infty})$ for every $f$ from any strongly dense subset $D^{*}$ of $X^{*}$ (where $X^{*}$ denotes the dual of $X$).

So suppose I have a Orlicz Space $L^{\Phi}(\mu)$ where $(\mathcal{X}, \mu)$ is a finite measure space and $\Phi$ is a Young function without the $\Delta_{2}$ property. In particular, I am interested in the Young function $\Phi(t) = e^{|t|} - 1$. Let $\|x_{n}\|$ be a sequence in $L^{\Phi}(\mu)$ satisfying condition i.): $\sup_{n} \|x_{n}\| < \infty$.

To finish showing weak convergence, I would then need to show that condition ii.) holds. Theorem 4 and its proof (see pages 121-122) give an approach on how to do this on the space $L^{1}(\mathcal{X},\mu)$. We define an appropriate set function as a limit: $\displaystyle \psi(B) = \lim_{n \rightarrow \infty} \int_{B} x_{n}(s) \mu(ds)$, use the Lebesgue-Nikodym theorem to show there exists $x_{\infty} \in L^{1}(\mathcal{X},\mu)$ such that $$ \lim_{n \rightarrow \infty} \int_{B} x_{n}(s) \mu(ds) = \int_{B} x_{\infty}(s) \mu(ds) \text{ for all } B, $$ and it follows that this equation holds for any simple function $g(s) = \sum_{j=1}^{k} \alpha_{j} \chi_{B_{j}}(s)$ ($\chi_{B_{j}}$ denoting the characteristic function): $$ \lim_{n \rightarrow \infty} \int_{B} x_{n}(s) g(s) \mu(ds) = \int_{B} x_{\infty}(s) g(s) \mu(ds). $$ Since simple functions are dense in $L^{\infty}(\mu) = (L^{1}(\mu))^{*}$ the second condition of Theorem 3 is satisfied.

It is this last step that seems to breakdown for Orlicz spaces without the $\Delta_{2}$ condition. The main hurdle is that, since $\Phi$ is not $\Delta_{2}$, its dual is not straightforwards. For $\Phi(t) = e^{|t|} - 1$, looking at Corollary 12 on page 124 of Rao & Ren's "Theory of Orlicz Spaces", elements of the dual are of the form $$ y^{*}(f) = \int f(s)h(s) \mu(ds) + \int f(s) \nu_{1} $$ where $h \in L^{\Psi}(\mu)$, the complementary Orlicz space, and $\nu_{1} \in B_{\Psi}$, where $B_{\Psi}$ is the set of finitely additive scalar set functions vanishing on $\mu$-null sets and with support contained in some $f \in L^{\Phi}(\mu) - \mathcal{M}^{\Phi}$. Note $\mathcal{M}^{\Phi}$ is the closed span of all step functions from $L^{\Phi}(\mu)$ and in the case of $\Phi(t) = e^{|t|}-1$, $\mathcal{M}^{\Phi} = M^{\Phi} = \left \{ f \in L^{\Phi}(\mu): \int \Phi(\alpha f) \mu < \infty \text{ for all } \alpha > 0 \right \}$ (see Rao & Ren, Chapter 3.4, Proposition 3, page 75).

Questions/observations:

1.) Are there known results for weak convergence involving non-$\Delta_{2}$ Orlilcz spaces? Ideally results that get around the more difficult dual spaces present under non-$\Delta_{2}$ conditions.

2.) Are there references (besides Rao & Ren) that deal with the dual spaces of non-$\Delta_{2}$ Orlicz spaces?

3.) Are there references detailing the role of the $\nu_{1}$ term: $\int f(s) \nu_{1} $ in the decomposition of an element of $(L^{\Phi}(\mu))^{*}$?

4.) I believe (but have not proved or found a reference) that the dual of $\Phi(t) = e^{|t|}-1$ is $\Psi(t) = t\log^{+}(t)$, where $\log^{+}(t) = \max(0,\log(t))$. Since this is a $\Delta_{2}$ function, simple functions are dense in $L^{\Psi}(\mu)$ (see "Stopping Times and Directed Processes" page 47 Corollary 2.1.18 or "Theory of Orlicz Spaces by Rao & Ren, page 77 Corollary 5). Is this somehow enough to prove weak convergence even with the presence of the $\int f(s) \nu_{1}$ term?