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?
Asked
Active
Viewed 207 times
1 Answers
0
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.