I am studying adjoint operators on non-$\Delta_{2}$ Orlicz spaces. Let $\Phi(t) = e^{|t|}-1$ be a Young function, noting it does not meet the $\Delta_{2}$ condition:
$$ \sup_{t \in [0,\infty)} \dfrac{\Phi(2t)}{\Phi(t)} < +\infty. $$
In the event the underlying measure space $(X,\mu)$ is finite, $x^{*} \in \left( L^{\Phi}(\mu) \right)^{*}$ is of the form: \begin{equation} x^{*}(f) = \int_{X} f g d\mu + \int_{X}f d\nu \end{equation} (see Rao & Ren "Theory of Orlicz Spaces" Chapter 4.2 Corollary 12, using the finiteness of the measure and $\Phi(t)$ continuous with $\Phi(t) > 0$ for $t > 0$). Note that $\Psi$ is the complementary Young function to $\Phi$, $g \in L^{\Psi}(\mu)$, and $\nu \in B_{\Psi}$ where $B_{\Psi}$ is the set of additive set functions with support in some $h \in \widetilde{L}^{\Phi} - \mathcal{M}_{\Phi}$ with $\widetilde{L}^{\Phi}$ the Orlicz class: \begin{equation} \widetilde{L}^{\Phi} = \left \{ f: \int_{X} \Phi(f) d\mu < +\infty \right \} \end{equation} and $\mathcal{M}_{\Phi}$ the closed linear span of all simple functions in $L^{\Phi}(\mu)$.
My question regards the second term: $\displaystyle \int_{X} f d\nu$. This can be decomposed into its component positive and negative parts: \begin{equation} \int_{X} f d\nu = \left( \int_{X} f^{+} d\nu^{+} - \int_{X} f^{+} d\nu^{-} \right) - \left( \int_{X} f^{-} d\nu^{+} - \int_{X} f^{-} d\nu^{-} \right). \end{equation} For simplicity take $f$ non-negative, then the component integrals can be expressed as: \begin{equation} \tag{1} \label{definition} \int_{X} f d\nu^{+,-} = \inf \left \{ \sum_{i=1}^{n} \inf_{s \text{ simple }} \| f \chi_{E_{i}} + s \|_{L^{\Phi}(\mu)} \nu^{+,-}(E_{i}): (E_{1}, \ldots, E_{n}) \in \mathcal{P} \right \} \end{equation} where $\mathcal{P}$ denotes a partition of $X$. This definition comes from Definition 3.7 in this paper.
$\textbf{Question: }$ How do we arrive at the above definition in Equation $\eqref{definition}$?
My initial thought had been that Equation $\eqref{definition}$ comes from applying the standard definition of a Lebesgue integral: \begin{equation} \int_{X} f d\nu = \sup \left \{ \int_{X} s d\nu: 0 \leq s \leq f, s \text{ simple} \right \} = \sup \left \{ \sum_{i=1}^{n} a_{i} \nu(E_{i}): a_{i} \geq 0, (E_{1}, \ldots, E_{n}) \in \mathcal{P} \right \}, \end{equation} but I am unsure why we are using $\inf_{s \text{ simple }} \| f \chi_{E_{i}} + s \|_{L^{\Phi}(\mu)}$ as the coefficient? My current thought is that it follows somehow from the fact that as an operator $\displaystyle \int_{X}f d\nu$ maps from $L^{\Phi}/\mathcal{M}_{\Phi}$ into $\mathbb{R}$. Any help/references appreciated.
