# Let ([0,1],R cap [0,1], lambda) be the standard probability space on [0,1] with λ, 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?

Geometric Interpretation of Independence of Random Variables on the Standard Probability Space
Let $\left(\left[0,1\right],\mathcal{R}\cap \left[0,1\right],\lambda \right)$ 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?
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Step 1
A collection of independent random variables (possibly infinite) defined on $\left(\mathcal{I},{\mathcal{B}}_{\mathcal{I}},\lambda \right)$, where $\mathcal{I}:=\left[0,1\right]$, can be constructed using a measure-preserving space-filling curve $\phi :\mathcal{I}\to \mathcal{I}×\mathcal{I}×\cdots$.
Step 2
The coordinate functions, $\left\{{\phi }_{i}\right\}$, are independent uniform random variables.