It is well known that $$\lim\limits_{r\rightarrow 0}\dfrac{\|f\chi_{B(x,r)}\|_{L_{p}(\mathbb{R}^{n})}}{\|\chi_{B(x,r)}\|_{L_{p}(\mathbb{R}^{n})}}=|f(x)|$$ for almost all $x\in\mathbb{R}^{n}$. Here $\chi_{B(x,r)}$ denotes the characteristic function of the open ball $B(x,r)$. I wonder whether there is an analogue of this property in Orlicz spaces, that is, $$\lim\limits_{r\rightarrow0}\dfrac{\|f\chi_{B(x,r)}\|_{L_{\Phi}(\mathbb{R}^{n})}}{\|\chi_{B(x,r)}\|_{L_{\Phi}(\mathbb{R}^{n})}}=|f(x)| \tag{$\ast$}$$ for almost all $x\in\mathbb{R}^{n}$ ?
Where $\Phi:[0,\infty)\to [0,\infty)$ is an increasing, continuous, convex function with $\Phi(0)=0$ and $$ \|f\|_{L_{\Phi}(\mathbb{R}^{n})}:=\inf\{\lambda>0:\int_{\mathbb{R}^{n}}\Phi\left(\frac{|f(x)|}{\lambda}\right)dx\leq 1\}. $$ It is a generalization of the $L_p$ norm. Indeed, if we take $\Phi(t)=t^p,\,1\leq p< \infty$ we get $\|f\|_{L_{\Phi}}=\|f\|_{L_{p}}$.
I think we should define a maximal function such that $$ M^{\Phi}f(x)=\sup_{r>0}\dfrac{\|f\chi_{B(x,r)}\|_{L_{\Phi}(\mathbb{R}^{n})}}{\|\chi_{B(x,r)}\|_{L_{\Phi}(\mathbb{R}^{n})}} $$ and obtain a weak type inequality for this operator. As an application of this we get the desired equation. Unfortunately, I couldn't do any of these operations.