## The Annals of Probability

### 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.

#### Article information

Source
Ann. Probab., Volume 7, Number 1 (1979), 166-169.

Dates
First available in Project Euclid: 19 April 2007

https://projecteuclid.org/euclid.aop/1176995159

Digital Object Identifier
doi:10.1214/aop/1176995159

Mathematical Reviews number (MathSciNet)
MR515824

Zentralblatt MATH identifier
0392.46020

JSTOR
Landers, D.; Rogge, L. A Functional Relationship between the Different $r$-means for Indicator Functions. Ann. Probab. 7 (1979), no. 1, 166--169. doi:10.1214/aop/1176995159. https://projecteuclid.org/euclid.aop/1176995159