I know that for any measure space $(\Omega,\Sigma,\mu)$ and any $\Sigma$-borel-measurble function $f\colon \Omega \to \mathbb{R}$ the integral
$$\int_\Omega |f(x)| \, d\mu(x)$$
is well definied. I just found in the book Stochastic Differential Equations of Oksendal the follwing definition (6th edition, Page 9):
Given a probability space $(\Omega,\Sigma,P)$ and a $\Sigma$-borel-measurble function $X \colon \Omega \to \mathbb{R}^n$ , then $\mu_X$ is a probability measure on $\mathbb{R}^n$ definied through
$$\mu_x(B) = P(X^{-1}(B))$$.
If $\int_\Omega |X(\omega)| \, dP(\omega) < \infty$ then the number
$$\int_\Omega X(\omega) \, dP(\omega) = \int_{\mathbb{R}^n} x d\mu_X(x) $$
is called the expecetation value.
However, I have never heard about the case of integrating a function $X\colon \Omega \to \mathbb{R}^n $. I could only assume that in this case $\int_\Omega X(x) \, \mu(dx)$ is a vector and $(\int_\Omega X(x) \, \mu(dx) )_i =\int_\Omega X_i(x) \, \mu(dx) $, but above it is clearly said that $\int_\Omega X(\omega) \, dP(\omega)$ is a number.
What is ment with $\int_\Omega |X(\omega)| \, dP(\omega)$? Did I miss something fundamental?
This is a misprint and your assumption is right. When $\displaystyle\int_\Omega \|X\|\mathrm dP$ is finite (as integral of a nonnegative real-valued function), the integral $\displaystyle\int_\Omega X\mathrm dP$ exists and is the point in $\mathbb R^n$ whose $i$th entry is $\displaystyle\int_\Omega X_i\mathrm dP$.