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

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?

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?