Why does there exist such dense set?

Question

Reference: Billingsley - Probability and measure p.379

Let $\mu$ be a Borel probability measure on $\mathbb{R}^k$.

Then, why does there exist a dense subset $D$ of $\mathbb{R}$ such that $\mu(\{x:x_i=d\})=0$ for $d\in D$ and $i=1,...,k$?

The author assumes this as trivial, but I don’t see why.. The only thing I can see now is that $D$ must be uncountable.

Thank you in advance.

Answer 1

Are you familiar with the result that for any random variable $X$, that $\{x\in \mathbb R:P(X=x)>0\}$ is countable?

If so, then consider the random variable $X_i$ on $(\mathbb R^n,\mathcal B(\mathbb R^n),\mu)$ defined by $(x_1,\dots,x_k)\mapsto x_i$. Then $S_i=\{x:P(X_i=x)>0\}$ is countable, and therefore $S=\bigcup_{i=1}^k S_i$ is as well. Simply take $D$ to be the complement of $S$; the complement of a countable set is always dense.