Please I need a reference where I can find a version of the "The reverse Lebesgue dominated convergence theorem" for the Orlicz spaces analogous to Theorem 1.2.7 in Semilinear Elliptic Equations for Beginners by Marino Badiale, Enrico Serra

Theorem 1.2.7. Let $\Omega\subseteq\mathbb R^N$ be open and let $\{u_k\}_k \subset L^p(\Omega)$, $p\in[1,+\infty]$, be a sequence such that $u_k\to u$ in $L^p(\Omega)$ as $k\to\infty$. Then there exists a subsequence $\{u_{k_j}\}_j$ and a function $v\in L^p(\Omega)$ such that:

  1. $u_{k_j(x)}\to u(x)$ a.e. in $\Omega$ as $j\to\infty$;
  2. for all $j$, $|u_{k_j}(x)|\le v(x)$ a.e. in $\Omega$.

This is Theorem 1.4 in Bennet and Sharpley's "Interpolation of Operators" (page 3). It actually holds for Banach spaces of functions equipped with what they call "function norms" and not only Orlicz spaces.

The full result is:

A map $\rho$ on the set of positive measurable functions on a measure space $(R,\mu)$ is a function norm if it fulfills

  1. $\rho(f)=0 \iff f=0\ \text{a.e.}$
  2. $\rho(af) = a\rho(f)$
  3. $\rho(f+g)\leq \rho(f)+\rho(g)$
  4. $0\leq g\leq f\ \text{a.e} \implies \rho(g)\leq\rho(f)$
  5. $0\leq f_n \nearrow f\ \text{a.e.}\implies \rho(f_n)\nearrow \rho(f)$
  6. $\mu(E)<\infty\implies \rho(\chi_E)<\infty$
  7. $\mu(E)<\infty\implies \int_E fd\mu\leq C_E\rho(f)$ for some $C_E$ independent of $f$.

Then it holds:

Theorem: Let $\rho$ be a function norm and $X$ be the space of measurable functions $f$ with $\rho(|f|)<\infty$. Then it holds

  1. With $\|f\|_X = \rho(|f|)$ it holds that $(X,\|\cdot\|_X)$ is a normed linear space.
  2. $X$ contains the linear combinations of characteristic functions of measurable sets.
  3. If $\|f_n-f\|_X\to 0$, then $f_n\to f$ in measure and, consequently, a subsequence of $f_n$ converges to $f$ pointwise a.e.

(see also Theorem 1.7 in the book which contains some more fact about this).

Later in the book they show that Orlicz norms are indeed function norms in their sense (Theorem 8.9 in Chapter 4.8).

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