Geometric Interpretation of Independence of Random Variables on the Standard Probability Space

Question

Let $([0,1],\mathcal{R}\cap[0,1],\lambda)$ be the standard probability space on $[0,1]$ with $\lambda$, the Lebesgue measure. What do independent random variables "look like" in this space? Is there an easy geometric approach to creating independent functions on this space?

Answer 1

A collection of independent random variables (possibly infinite) defined on $(\mathcal{I},\mathcal{B}_{\mathcal{I}},\lambda)$, where $\mathcal{I}:=[0,1]$, can be constructed using a measure-preserving space-filling curve $\varphi:\mathcal{I}\to \mathcal{I}\times \mathcal{I}\times\cdots$. The coordinate functions, $\{\varphi_i\}$, are independent uniform random variables. Examples of such maps can be found in this book.