The Hausdorff Metric of $\sigma$-Fields and the Value of Information

Abstract

Using a result by Landers and Rogge, it is shown that the Hausdorff metric of $\sigma$-fields is uniformly equivalent to the metric induced by the Hausdorff distance between sets of measurable functions. An application is given to the continuity of the value of information with respect to the Hausdorff metric of $\sigma$-fields.