Constituting Atoms of a $σ$ Algebra via Its Generator

Jinshan Zhang

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arXiv · 0711.2400

Constituting Atoms of a $σ$ Algebra via Its Generator

Jinshan Zhang

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Abstract

To constitute atoms of a $σ$ algebra is not a easy task due to the large number of its elements. However, determining them via generators seems a feasible and simple way since most $σ$ algebras are generated by their smaller proper subsets. Precisely, under some conditions each atom of a $σ$ algebra equals the intersection of the elements containing a point of the atom in the generator. In this paper, a very weak sufficient condition for determining atoms by the generator is presented. The condition, though not being a necessary one, is shown to be almost the weakest one in the sense that it can hardly be improved.

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Jinshan Zhang. 2008-12-11. Constituting Atoms of a $σ$ Algebra via Its Generator. https://arxiv.org/abs/0711.2400

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