I would like to find a referenceable source of the following cool technique to get an explicit bijection $h:[0,1)\rightarrow [0,1)^{\mathbb{N}}$ that is measurable and has measurable inverse:
Cool technique developed in [?]: Write $x \in [0,1)$ in its unique binary expansion that does not contain an infinite tail of 1s and then parse that expansion by: $$x = 0.w_1w_2w_3...$$ where $w_i \in \{0, 10, 110, 1110, ...\}$. Then define $h:[0,1)\rightarrow[0,1)^{\mathbb{N}}$ by defining each component function $h_i(x)$ for $i \in \mathbb{N}$ as: $$ h_i(x) = 0.w_{\phi(i,1)}w_{\phi(i,2)}w_{\phi(i,3)}...$$ where $\phi:\mathbb{N}^2\rightarrow\mathbb{N}$ is any bijection. Clearly the given expansion for each $h_i(x)$ does not contain an infinite tail of 1s since each word $w_i$ ends with 0.
Reference note: I remember reading about this on either mathoverflow or stackexchange, which is how I know about the technique, but I cannot find the link again! Chapter 13 in Dudley's book Real Analysis and Probability treats this topic using single-bit binary expansions rather than the variable-bit expansion in the above technique. Durrett's book and Kallenberg's book both cite Dudley. So these three references have the main result on existence of a bijection, but do not have the explicit bijection given above.