Continuum hypothesis and non measurable set

Question

This is from Chap 8 of Real and Complex analysis of Rudin.

The author does not present a proof (using the continuum hypothesis) for the existence of the function $j$. Where can I find such a proof?

Answer 1

The continuum hypothesis says $2^{\aleph_0} = \aleph_1$, right?

The cardinal of $[0,1]$ is $2^{\aleph_0}$.

By definition, $\aleph_1$ is the cardinal of the least uncountable ordinal. That is, a well-ordered set that is uncountable, and each initial segement is countable.

The last step: if two sets have the same cardinal, then there is a one-to-one mapping from one onto the other.