Extension of The Best Approximation Operator in Orlicz Spaces

Abstract

Let $(\Omega{},\mathcal{A},\mu{})$ be a probability space and $\mathcal{L}\subset{}\mathcal{A}$ a sub-$\sigma{}$-lattice of the $\sigma{}$-algebra $\mathcal{A}$. We study an extension of the best $\phi{}$-approximation operator from an Orlicz space ${L}^{\phi{}}$ to the space ${L}^{{\phi{}}^{\prime{}}}$, where ${\phi{}}^{\prime{}}$ denotes the derivative of the convex, but not necessarily a strictly convex function $\phi{}$. We obtain convergence results when a sequence of $\sigma{}$-algebras ${\mathcal{B}}_{\text{n}}$ converges to ${\mathcal{B}}_{\infty{}}$ in a suitable way.