Applying compactness of subsets

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This is a survey article about applying the concept/definition/theorem: compactness

This is a survey article explaining how we can prove results about non-compact spaces, using compactness-type arguments on subsets. Specifically, we shall be interested in the following notions:

  • Locally compact Hausdorff space: A topological space where every point is contained in an open subset whose closure is compact.
  • Sigma-compact space: A topological space with a countable collection of compact subsets whose interiors cover the whole space.

Proving results for regular measures

One of the powerful ways of applying compactness of subsets is when we need to prove results for inner regular Borel measures. An inner regular Borel measure is a measure defined on the Borel subsets, such that for any open subset, the measure of the subset is the supremum of the measures of all compact subsets contained inside it. For an inner regular measure, we can prove bounds on the measure of an open subset by proving a bound on the measure of any compact subset contained inside it.

An example is proving that certain maximal operators are of weak (1,1)-type: for instance, the maximal operator which sends a function f \in L^1(\Omega) (with \Omega a subset of Euclidean space) to Mf, the maximal function of f, is of weak (1,1)-type with the constant being 3^n. The proof of this relies on considering the open subsets of the form: x : Mf(x) > \lambda, and showing that the measure of any compact subset contained in such a set, is bounded from above.