# Showing that there do not exist uncountably many independent, non-constant random variables on $([0,1],\mathcal{B},\lambda)$.

I have this problem in my assignment:

Show that there do not exist uncountably many independent, non-constant random variables on $([0,1],\mathcal{B},\lambda)$, where $\lambda$ is the Lebesgue measure on the Borel $\sigma$-algebra $\mathcal{B}$ of $[0,1]$.

Assume that $(X_i)_{i \in I}$ are independent random variables. Let $Y_i := X_i \cdot 1_{|X_i|\leq C_i}$, where $C_i>0$ is chosen so large that $Y_i \not\equiv c_i$ for some constant $c_i$. This is possible since the $X_i$ are non-constant.
Then the $(Y_i - \Bbb{E}(Y_i))_i$ form a family of independent and hence orthogonal random variables. Note that the $Y_i$ are bounded and thus contained in $L^2$.
But the separable (!) space $L^2 ([0,1],\lambda)$ can only contain countably many elements which are mutually orthogonal. Hence, $I$ is countable.
• Does this also imply that there do not exist uncountably many orthogonal, non-constant random variables on $([0,1], \mathcal{B}, \lambda)$? – gigalord Jan 9 at 21:01