Definition: The Borel sigma-algebra of a set S is the smallest sigma-algebra of S
that contains all of the open balls in S. Any element of a Borel
sigma-algebra is a Borel Set.
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Example: The set B1 is the Borel sigma-algebra of the real line, and thus contains every open interval.
Example: Consider a filled circle in the unit square. It can be constructed by a countable number of non-overlapping open rectangles (since a series of such rectangles can be defined that would cover every point in the circle but no point outside of it. Therefore it is in the smallest sigma-algebra of open subsets of the unit square.(Econterms)
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