Convergence of Conditional $p$-Means Given a $\sigma$-Lattice

Abstract

Let $\mathbf{P}_n \mid \mathscr{A}, n \in \mathbb{N}$, be a sequence of probability measures converging in total variation to the probability measure $\mathbf{P} \mid \mathscr{A}$ and $\mathscr{C}_n \subset \mathscr{A}, n \in \mathbb{N}$, be a sequence of $\sigma$-lattices converging increasing or decreasing to the $\sigma$-lattice $\mathscr{C}$. Then for every uniformly bounded sequence $f_n, n \in \mathbb{N}$, converging to $f$ in $\mathbf{P}$-measure we show in this paper that the conditional $p$-mean $\mathbf{P}_n^\mathscr{C}k f_j$ converge to $\mathbf{P}^\mathscr{C}f$ in $\mathbf{P}$-measure if $n, k, j$ tends to infinity. The methods used in this paper are completely different from those used to prove the corresponding result for $\sigma$-fields instead of $\sigma$-lattices.