Spectral Theory
Borel sets are a collection of sets that can be formed from open intervals through countable unions, countable intersections, and relative complements. They play a crucial role in measure theory and topology, providing a framework to define measurable spaces and establish concepts like continuity and convergence. The significance of Borel sets extends to spectral measures and projection-valued measures, as they are used to categorize subsets of the spectrum of an operator, which is essential for understanding the spectral properties of operators in functional analysis.
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