Convergence of $L_{p}$ approximations as $p\rightarrow \infty$

Abstract:

Let $(\Omega , \mathcal {A}, \mu )$ be a probability space and let $\mathcal {B}$ be a subsigma-algebra of $\mathcal {A}$. Let $A = {L_\infty }(\Omega , \mathcal {A}], \mu )$ and let $B = {L_\infty }(\Omega ,\mathcal {B},\mu )$. Let $f \in A$, and for $1 < p < \infty$, let ${f_p}$ denote the best ${L_p}$ approximation to $f$ by elements of ${L_p}(\Omega ,\mathcal {B},\mu )$. It is shown that ${\lim _{p \to \infty }}{f_p}$ exists a.e. The function ${f_\infty }$ defined by ${f_\infty }(x) = {\lim _{p \to \infty }}{f_p}(x)$ is a best ${L_\infty }$ approximation to $f$ by elements of $B:||f - f_\infty ||_\infty = \inf \{ ||f - g||_\infty ; g \in B \}$. Indeed, ${f_\infty }$ is a best best ${L_\infty }$ approximation to $f$ by elements of $B$ in the sense that for each $E \in \mathcal {B}$ the restriction, ${f_\infty }|E$, of ${f_\infty }$ to $E$ is a best ${L_\infty }$ approximation to the restriction, $f|E$, of $f$ to $E$. Since there is at most one best best ${L_\infty }$ approximation to $f$,${f_\infty }$, is the best best ${L_\infty }$ approximation to $f$ by elements of $B$.