Approximation with Independent Variables

Given a square integrable m-dimensional random variable $X$ on a probability space $(Ω.\mathcal F,\Pr)$ and a sub sigma algebra $\mathcal A$, we show that there exists another m-dimensional random variable $Y$, independent of $\mathcal A$ and minimising the $L^2$ distance to $X$. Such results have an importance to fairness and bias reduction in Artificial Intelligence, Machine Learning and Network Theory.

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