Abstract
This paper investigates smoothness properties of probability measures on lattices which imply regularity, and then considers weaker versions of regularity; in particular, weakly regular, vaguely regular, and slightly regular. They are derived from commonly used outer measures, and we analyze them mainly for the case of I(ℒ) or for those elements of I(ℒ) with added smoothness conditions.
Highlights
Let X be an arbitrary set and a lattice of subsets of X
E.g., tz’ #" on if and only if # G J(): those # G I() such
We adhere to standard lattice and measure terminology which will be used throughout the paper and review some of this in section two for the reader’s convenience
Summary
Let X be an arbitrary set and a lattice of subsets of X. A((2) denotes the algebra generated by and I() those non-trivial zero-one valued finitely additive measures on. Conditions for #’ #" or # tz" on are investigated. This leads to a consideration of weak notions of regularity, two of which can be expressed in terms of #’ and #". We show that if la J() and if is complement generated # is weakly regular. Combining these results gives conditions for certain measures to be regular. We adhere to standard lattice and measure terminology which will be used throughout the paper We adhere to standard lattice and measure terminology which will be used throughout the paper (see e.g. [1,4,5]) and review some of this in section two for the reader’s convenience
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