If $X \sim F$ where the support of $X$ is $\mathbb{R}^p$. So, $X = (X\_1, X\_2, \dots, X\_p)$. Then say I assume $X$ has $k$ finite moments. When $p = 1$, I know that that means 
$$\int\_{\mathbb{R}} x^k\, f(x)\, dx < \infty, $$
where $f(x)$ is the associated density of $F$. What the mathematical equivalent of assuming $X$ has $k$ finite moments when $p > 1$?

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In [this](http://www.deaneyang.com/papers/affine_moment.pdf) link, on page 2, the authors define the $k$th moment as
$$E\|X\|^k = \int \|X\|^k f(x) \, dx, $$
where $\| \cdot\|$ is the Euclidean norm.

Glen\_b's answer [here](https://stats.stackexchange.com/questions/188389/what-is-the-moment-of-a-joint-random-variable) suggests that the $k$th moment would be
$$\int x\_1^kx\_2^k \dots x\_p^k \, f(x) dx. $$

Does assuming one to be finite imply the other is finite?