Infinite Moment of a measure. i found the following definition:
For a distribution $\mathrm{Q}$ on $\mathrm{R}$, the $p$-th moment, $p \in(0, \infty)$, is defined by
$$
|\mathrm{Q}|_{p}:=\left(\int_{\mathbb{R}}|y|^{p} d \mathrm{Q}(y)\right)^{1 / p}
$$
Moreover, its $\infty$-moment is defined by $|\mathrm{Q}|_{\infty}:=\sup |\operatorname{supp} Q|$.
My question is now how to understand the supremum of the support? Does anyone know what is meant here?
 A: The support $S$ of a distribution function $Q$ is the set of all real numbers $a$ such that $Q(a-\epsilon, a+\epsilon) >0$ for every $\epsilon >0$. The supremum of  $\{|a|: a\in S\}$ is $|Q|_{\infty}$.
A: In the context of your OP,
$$|\operatorname{supp}(Q)|=\{|x|: x\in\operatorname{supp}(Q)\}$$
The support of a measure $Q$ on $\mathbb{R}$ is defined as $$\operatorname{supp}(Q)=\{x\in \mathbb{R}: Q((x-\varepsilon,x+\varepsilon)>0,\,\forall \,\varepsilon>0\}$$
Observe that if $X:\mathbb{R}\rightarrow\mathbb{R}$ is the identity map, i.e. $X(x)=x$, then for $0<p<\infty$ $$|Q|_p=\|X\|_p=\Big(\int_{\mathbb{R}}|X|^p\,dQ\Big)^{1/p},$$
that is the familiar $L_p(Q)$ norm of the function $X$.
Recall that the $L_\infty(Q)$ norm of a real measurable function $Y$ on $\mathbb{R}$ is defined as
$$\|Y\|_\infty=\inf\{\alpha>0: Q(\{|Y|>\alpha\})=0\}$$
It is easy to check that the $L_\infty(Q)$ norm of $X$, $\|X\|_\infty$, coincides with the definition $|Q|_\infty$ in the OP, that is
$$\|X\|_\infty=\inf\{\alpha>0: Q(\{|X|>\alpha\})=0\}=\sup\{|x|: x\in\operatorname{supp}(Q)\}=|Q|_\infty$$.
