What is the 0-norm?

On $\mathbb{R}^n$ and $p\ge 1$ the $p$-norm is defined as $$\|x\|_p=\left ( \sum _{j=1} ^n |x_j| ^p \right ) ^{1/p}$$
and there is the $\infty$-norm which is $\|x\|_\infty=\max _j |x_j|$. It’s called the $\infty$ norm because it is the limit of $\|\cdot\|_p$ for $p\to \infty$.

Now we can use the definition above for $p<1$ as well and define a $p$-“norm” for these $p$. The triangle inequality is not satisfied, but I will use the term “norm” nonetheless. For $p\to 0$ the limit of $\|x\|_p$ is obviously $\infty$ if there are at least two nonzero entries in $x$, but if we use the following modified definition
$$\|x\|_p=\left ( \frac{1}{n} \sum _{j=1} ^n |x_j| ^p \right ) ^{1/p}$$
then this should have a limit for $p\to 0$, which should be called 0-norm. What is this limit?

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