Let $g(f)$, $S(f)$, $g_\lambda^*(f)$ be the Littlewood-Paley $g$ function, 
Lusin area function, and  Littlewood-Paley $g_\lambda^*$ function of $f$, 
respectively. In 1990 Chen Jiecheng and Wang Silei showed that if, 
for a $\mathrm{BMO}$ function $f$, one of the above functions is finite 
for a single point in $\mathbb R^n$, then it is finite a.e. on 
$\mathbb R^n$, and $\mathrm{BMO}$ boundedness holds. Recently, 
Sun Yongzhong extended this result to the case of Campanato spaces 
(i.e. Morrey spaces, $\mathrm{BMO}$, and \Lipschitz spaces). We improve 
his $g_\lambda^*$ result further. His assumption is $\lambda>3+2/n$. We 
show this is relaxed to $\lambda>\max(1,{2/p})$ $(-{n/p} \le \alpha < 0)$, 
$\lambda>1$ $(0 \le \alpha <{1/2})$, and $\lambda>1+ {2\alpha/n}$ 
$({1/2} \le \alpha <1)$. 
We also treat generalized Marcinkiewicz functions $\mu^{\rho}(f)$, 
$\mu_S^{\rho}(f)$ and $\mu_\lambda^{\ast, \rho}(f)$.