It's quite common to ask the size of a set in terms of its cardinality. In the case of the vector space $\mathbb{R}$ over $\mathbb{Q}$, to ask for the cardinality of a basis of $\mathbb{R}$ over $\mathbb{Q}$ or even to give it a subset of it in which it is countable. In my sense, I am curious to know how common a vector $\boldsymbol{v} \in \mathbb{R}^{n}$ has coordinates that "randomly fall" in a linearly independent set of $\mathbb{R}$ over $\mathbb{Q}$. Hence, this question can be thought of as:
What is the largest Lebesgue measure a basis of $\mathbb{R}$ over $\mathbb{Q}$?
If a basis of $\mathbb{R}$ over $\mathbb{Q}$ has a positive measure, it's reasonable to say that it's "common" to have a $\boldsymbol{v} \in \mathbb{R}^{n}$ that "randomly fall" in a linearly independent set of vectors.
