Convergence Results for Approximation with independent Variables

For a square integrable $m$-dimensional random variable $X$ on a probability space $(\mathrm{\Omega },\text{Fc},\mathrm{Pr})$ and a sub sigma algebra $\text{Ac}$, we show that there is a constructive way to represent $X-\text{Er}[X\mid \text{Ac}]$ as the sum of a series of variables that are independent of $\text{Ac}$.