Extension of The Best Approximation Operator in Orlicz Spaces

Abstract

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