# Definition of $L^p$-spaces of random variables

I'm reading Bernt Oksendal's "Stochasticc Differential Equations" and got confused at the definition of $L^p$-space of random variables.

In chapter 2, page 9 (sixth edition) it says (consider a given probability space $(\Omega, \mathscr{F}, P)$):

If $X:\Omega \rightarrow \mathbb{R}^n$ is a random variable and $p\in[1,\infty)$ is a constant we define the $L^p$-norm of $X$, $\| X \|_p$, by $$\|X\|_p = \|X\|_{L^p(P)} = \left(\int_{\Omega}|X(\omega)|^p dP(\omega)\right)^\frac{1}{p}$$

My questions is, how is the $|X(\omega)|$ defined?

$X$ itself is a mapping (function) from $\Omega$ to $\mathbb{R}^n$, since $\omega \in \Omega$, $X(\omega) \in \mathbb{R}^n$.

A norm in $\mathbb{R}^n$ itself could be defined in many many ways, such as $L^p$ norm.

Is the $|X(\omega)|$ here free to be defined by any norm in $\mathbb{R}^n$?

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A norm in $\mathbb{R}^n$ itself could be defined in many many ways, such as $L^p$ norm.
True, but the resulting $L^p$ space is always the same set, since all the norms on $\mathbb{R}^n$ are equivalent. Simply, for $X$ in $L^p$, $\|X\|_p$ will not be the same.
Is the $|X(\omega)|$ here free to be defined by any norm in $\mathbb{R}^n$?
what do you mean by all the norms on $\mathbb{R}^n$ are equivalent? –  athos Sep 21 '13 at 16:56