A Functional Relationship between the Different $r$-means for Indicator Functions

Abstract

Let $P$ be a probability measure defined on a $\sigma$-field $\mathscr{F}$ over $\Omega$. Let $\mathfrak{L}\subset \mathscr{F}$ be a $\sigma$-lattice and $r > 1$. For each $A \in \mathscr{F}$ denote by $P_r(A/\mathfrak{L})$ the unique nearest point projection of $1_A$ onto the closed convex subspace of all "$\mathfrak{L}$-measurable" equivalence-classes of $L_r(\Omega, \mathscr{F}, P)$. It is shown that there exists a functional relationship between $P_r(A/\mathfrak{L})$ and $P_2(A/\mathfrak{L})$ of the form $$P_r(A/\mathfrak{L}) = \varphi(P_2(A/\mathfrak{L}))$$ where the function $\varphi$ depends only on $r$ but not on $A, P$ or $\mathfrak{L}$. This relationship is applied to the theory of sufficiency.