Let $(\Omega, \mathcal{ F}, \mathbf{ P})$ be a probability space, $\xi$ be a random variable on $(\Omega, \mathcal{ F}, \mathbf{ P})$, $\mathcal{ G}$ be a sub-$\sigma$-algebra of $\mathcal{ F}$, and let $\mathbf{ E}^\mathcal{ G} = \mathbf{ E}(\cdot | \mathcal{ G})$ be the corresponding conditional expectation operator. We obtain sharp estimates for the moments of $\xi - \mathbf{ E}^\mathcal{ G}\xi$ in terms of the moments of $\xi$. This allows us to find the optimal constant in the bounded compact approximation property of $L^p([0, 1])$, $1 < p < \infty$.
Categories: math.PR math.FA