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Does the following concept have a standard name? Has it been studied?

A sub-$\sigma$-algebra that has a null-set "surrounding" any null-set of the underlying $\sigma$-algebra.

More precisely, let $\left(\Omega, \mathcal{A}, \mu\right)$ be a measure space and define $\beta$ to be the collection of those sub-$\sigma$-algebras of $\mathcal{A}$ that are characterized by the following property: $\mathcal{B}\in\beta$ iff whenever $E\in\mathcal{A}$ with $\mu(E)=0$, there's some $F\in\mathcal{B}$ with $\mu(F)=0$ such that $E\subseteq F$.

As an example, the Borel field on the real line is a member of $\beta$ w.r.t. the Lebesgue field on the real line.

The utility of such a concept is, for instance, in that if $f_n$ is a sequence of $\mathcal{B}$-measurable functions that converge to a function $g$ $\mu$-almost everywhere, then $g$ is $\mathcal{B}$-measurable.

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up vote 1 down vote accepted

In analogy with the "Lebesgue measure is Borel regular", perhaps something like $\beta$ is the collection of all sub-$\sigma$-algebras $\mathcal{B}$ of $\mathcal{A}$ such that $\mathcal{A}$ is "$\mathcal{B}$-regular". However, since the $\mu$-measure zero sets are not specified when you only mention the $\sigma$-algebras, it would probably be better to use terms that explicitly mention the measures and not just the $\sigma$-algebras. Note that a measure implicitly includes its underlying $\sigma$-algebra as part of the definition of the measure.

See, for example, how Dan Brigham handles these issues on pp. 93-96 of his May 2011 University of Missouri M.A. Thesis Quasi-Metric Geometry: Smoothness and Convergence Results.

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Thank you, Dave. –  Evan Aad Mar 7 '13 at 6:28

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