On convergence in the Orlicz space as a ranked space (Q2788943)

Let \(p: [0,\infty)\to[0,\infty)\) be a nondecreasing right continuous function with \(\lim_{u\to\infty}p(u)=\infty\). The function \(\varphi\) defined by NEWLINE\[NEWLINE\varphi(t)= \int^t_0 p(u)\,duNEWLINE\]NEWLINE is convex and called an \(N\)-function. For \(\alpha>0\), the set \(\varphi(\alpha L)\) of functions which are Lebesgue measurable on \([-\pi,\pi]\) is defined by NEWLINE\[NEWLINE\varphi(\alpha L)= \Biggl\{\text{measurable }f: \int^\pi_{-\pi} \varphi(\alpha|f(x)|) \,dx<\infty\Biggr\},NEWLINE\]NEWLINE and the set \(V(f;\alpha, \varepsilon)\) is defined for \(\varepsilon>0\) by NEWLINE\[NEWLINEV(f,\alpha,\varepsilon)= \Biggl\{g\in\varphi(\alpha L): \int^\pi_{-\pi} \varphi(\alpha|f(x)-g(x)|) \, dx< \varepsilon\Biggr\}.NEWLINE\]NEWLINE The Orlicz space \(L^*_\varphi\) is defined by \(L^*_\varphi= \bigcup\{\varphi(\alpha L),\, \alpha>0\}\), and the Luxemburg-Nakano (L-N) norm \(\| f\|_{(\varphi)}\) is defined by NEWLINE\[NEWLINE\| f\|_{(\varphi)}= \text{inf}\Biggl\{\lambda>0: \int^\pi_{-\pi} \varphi(\lambda^{-1}|f(x)|)\,dx< \infty\Biggr\}.NEWLINE\]NEWLINE The Orlicz space \(L^*_\varphi\) is considered to be a ranked space since each \(f\) in \(L^*_\varphi\) has pre-neighbourhoods of the form \(V(f;\alpha_n, 2^{-n})\), \(n= 1,2,\dots\).NEWLINENEWLINE The main theorems of this paper appear to relate convergence in the L-N norm to ortho-convergence (or \(r\)-convergence) in terms of the neighbourhoods. In particular, if \(\{V_n(f_n; \alpha_n, \varepsilon_n)\), \(n= 1, 2,\dots\}\) is a fundamental sequence of pre-neighbourhoods, so that \(V_m\subset V_n\), \(0<\varepsilon_m< \varepsilon_n\), for \(n<m\), and \(\varepsilon_n\to 0\) as \(n\to\infty\), then any sequence \(\{g_n: g_n\in V_n\}\) is a Cauchy sequence in measure and is ortho-convergent to \(f\) in \(\varphi(\alpha_n L/2)\), \(n= 1,2,\dots\).