It is shown that the Hausdorff-metric of $\sigma$-fields--which plays an important role for uniform martingale theorems--has a surprising "additivity" property. For example this property can be used to obtain a sharpened version of a uniform inequality for conditional expectations.
"An Inequality for the Hausdorff-Metric of $\sigma$-Fields." Ann. Probab. 14 (2) 724 - 730, April, 1986. https://doi.org/10.1214/aop/1176992541