Distances Between σ-Fields on a Probability Space

Abstract

For a probability space (Ω,ℱ,P) and two sub-σ-fields $\mathcal{A},\mathcal{B}\subset\mathcal{F}$ we consider two natural distances: $\rho(\mathcal{B},\mathcal{A})=\sup_{B\in\mathcal{B}}\inf_{A\in\mathcal{A}}P(A\triangle B)$ and $\overline{\rho}(\mathcal{B},\mathcal{A})=\sup_{B\in\mathcal{B}}\inf_{A\in\mathcal{A},A\supset B}P(A\setminus B)$ . We investigate basic properties of these distances. In particular we show that if a distance (ρ or $\overline{\rho}$ ) from ℬ to $\mathcal{A}$ is small then there exists Z∈ℱ with small P(Z), such that for every B∈ℬ there exists $A\in\mathcal{A}$ such that B∖Z and A∖Z differ by a set of probability zero. This improves results of Neveu (Ann. Math. Stat. 43(4):1369–1371, [1972]), Jajte and Paszkiewicz (Probab. Math. Stat. 19(1):181–201, [1999]).