A function $\varphi:\mathbb{R}\to[0,\infty]$ is called a Young function if $\varphi $ is convex, even,and left continuous with $\varphi(0)=0 $, also let $G$ denote a locally compact group with a left Haar measure $\lambda$. and $$M^{\varphi}(G)=\{f :G\to \mathbb{C}: f\;\text{is measurable },\;\int_G \varphi(a|f(x)|) \,\mathrm d\lambda(x)<\infty\; \;\text{for all }a>0\}$$ I want to prove the following lemma
LemmaLet $\varphi$be a strictly increasing, continuous Young function the following statements are equivalent
- $M^{\varphi}(G)\subseteq L^{\infty}(G)$
2.$G$ is discrete
My Solution:
I've done a proof for $2\to1:$
$1\to2:$ Suppose $G$ is not discrete. Then there exist a relatively compact neighborhood $K$ and a sequence $(K_n)$ of pairwise disjoint open subset of $K$ such that $$\lambda(K_n)\varphi(n^2)<\lambda(K)n^{-2}$$ for all $n\geq1.$ Define the function $f$ on $G$ by $$f(x)=\sum_{n=1}^{\infty}n\chi_{K_n}(x)$$ but I have a problem to prove $f\in M^{\varphi}(G)$ and $\lambda(K_n)\varphi(n^2)<\lambda(K)n^{-2}$
Any idea or hint for a reference is welcome!