Hello I need help understanding a proof. The proof is from Rudin's Real and Complex Analysis. The proof is supposed to address the question: Is every Lebesgue measurable set a Borel set?
Let $c$ be the cardinality of the continuum (the real line or, equivalently, the collection of all sets of integers). We know that $\mathbb R^k$ has a countable base (open balls with rational radii and with centers in some countable dense subset of $\mathbb R^k$) and that $\mathcal B_k$ (the collection of all Borel sets of $\mathbb R^k$) is the $\sigma$-algebra generated by this base. It follows from this that $\mathcal B_k$ has cardinality $c$. On the other hand, there exist Cantor sets $E\subset\mathbb R^1$ with $m(E)=0$ (where $m$ is a measure). The completeness of $m$ implies that each of the $2^c$ subsets of $E$ is Lebesgue measurable. Since $2^c>c$, most subsets of $E$ are not Borel sets
I do not understand why the $2^c$ is needed. Neither do I understand how it is so suddenly introduced. Where does the $2^c$ come from? Also, how does the fact that $2^c>c$ show that most subsets of $E$ are not Borel sets? What is the relation between cardinality and Borel sets?