I was wondering this question today, but I don't really have the machinery to answer it, and I'm not sure how to search for such an answer. Here's the set up of the problem:
What is the order of infinity of the set of vectors whose components sum to $0$. Put differently, take a vector $x \in \mathbb{R}^n$. If $A= \{x | \mathbf{1} \cdot x =0 \} $ then what is the $size(A)$? And how does it compare to the order of infinity of the set of $x' \in \mathbb{R}^n$? Is $x$ dense in $x'$? If there was a uniform distribution across all of $\mathbb{R}^n$, what would be the probability of selecting such an $x$? Intuitively, this last question seems to be related to the cardinality of $x$ but still somewhat different.
The cardinality is the same as $\Bbb R^{n-1}$, which is the size of the continuum for $n \gt 1$. You can choose the first $n-1$ coordinates at will, then have to choose the last one to make the sum zero. It forms an $n-1$ dimensional surface in $R^n$ with $0 \ n-$ dimensional volume. From the measure point of view the chance of getting one of these vectors is zero just like the chance of picking a specific real.