Let $\Omega$ be a set, $\mathbb{P}$ a probability measure on $\Omega$, and $\Sigma_0\subset \Sigma$ be finite $\sigma$-algebras on $\Omega$ on which $\mathbb{P}$ is defined. Let $X$ be a Banach space, $1\leqslant p<\infty$, and let $L_p(X)$ denote the Banach space of (equivalence classes) of Bochner $p$-integrable $X$ valued functions on $\Omega$ strongly measurable with respect to $\Sigma$. Let $L_p(X, \Sigma_0)$ be defined similarly. Let $Z$ denote the subspace of $L_p(X, \Sigma)$ consisting of those $f\in L_p(X, \Sigma)$ such that $\mathbb{E}(f|\Sigma_0)=0$. Then $f\mapsto f-\mathbb{E}(f|\Sigma_0)$ is a projection onto $Z$. Standard facts yield that $f\mapsto \mathbb{E}(f|\Sigma_0)$ is a norm $1$ projection of $L_p(X, \Sigma)$ onto $L_p(X, \Sigma_0)$, so the complementary projection $f\mapsto f-\mathbb{E}(f|\Sigma_0)$ of $L_p(X, \Sigma)$ onto $Z$ is norm at most $2$.

What is the exact norm of $f\mapsto f-\mathbb{E}(f|\Sigma_0)$?


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