Borelmengen

Borelmengen, or Borel sets, are a fundamental concept in measure theory and analysis. In a topological space X, the Borel σ-algebra B(X) is the smallest σ-algebra that contains every open set. A Borel set is any member of this σ-algebra. This makes Borel sets the natural class of sets for defining and studying measurable functions and probability on spaces with a topology.

Key properties include closure under countable union, countable intersection, and complementation. Consequently, all open and closed sets are Borel, and more complex sets such as F_sigma sets (countable unions of closed sets) and G_delta sets (countable intersections of open sets) are Borel as well. In a metric space, Borel sets can also be generated by a countable base: for R^n with its usual topology, the Borel σ-algebra is generated by the open intervals (or, equivalently, by open balls with rational centers and radii).

Examples illustrate the scope: every open set and every closed set in R is Borel; the rationals Q are a countable union of singletons, each of which is closed, so Q is an F_sigma set and thus Borel; the irrationals R\Q form a G_delta set. The Cantor set is closed and hence Borel.

Significance extends beyond topology: Borel sets provide the natural domain for defining the Borel measurable functions, which are central in probability and analysis. They also serve as a gateway to Lebesgue measure, since Lebesgue measure restricted to Borel sets is a fundamental example of a measure. In general topological spaces, Borel sets help compare different notions of measurability and play a central role in descriptive set theory.