Uniform Inequalities for Conditional $p$-Means Given $\sigma$-Lattices

Abstract

Let $(X, \mathscr{F}, P)$ be a probability space and $\mathscr{A}$ a sub-$\sigma$-lattice of $\mathscr{F}: \mathscr{A}$ is closed under countable union and countable intersection and contains $X$ and $\varnothing$. Let $P^\mathscr{A}$ denote conditional expectation given $\mathscr{A}$ (Barlow and coworkers, 1972), and for fixed $p \geqq 1$ let $M^\mathscr{A}$ denote conditional $p$-mean given $\mathscr{A}$ (Brunk and Johansen, 1970). Rogge showed (1974) that for $\sigma$-fields $\mathscr{A}$ and $\mathscr{B}, \sup \{\|P^\mathscr{A} f - P^\mathscr{B} f\|_2: 0 \leqq f \leqq 1\} \leqq 2\{\delta(\mathscr{A}, \mathscr{B})\lbrack 1- \delta(\mathscr{A}, \mathscr{B}) \rbrack\}^{\frac{1}{2}}$, where $\delta(\mathscr{A}, \mathscr{B}) \equiv \max\{\sup_{A\epsilon\mathscr{A}} \inf_{B\epsilon\mathscr{B}} P(A \triangle B), \sup_{B\epsilon\mathscr{B}} \inf_{A\epsilon\mathscr{A}} P(A \triangle B)\}:$ and that for $p \geqq 1$ the convergence to 0 of $\|P^\mathscr{F} n f - P^\mathscr{F} \infty f\|_p$ is uniform for $|f| \leqq 1$ if $\delta(\mathscr{F}_n, \mathscr{F}_\infty) \rightarrow 0$. In the present paper an inequality for conditional $p$-means given $\sigma$-lattices similar to Rogge's is obtained for $p \geqq 2$ and is applied to obtain uniformity of convergence to 0 of $\|M^\mathscr{A}_n f - M^\mathscr{A} \infty f\|_p$ when $\delta(\mathscr{A}_n, \mathscr{A}_\infty) \rightarrow 0$.