PUMaC 2024 · 加试 · 第 3 题

PUMaC 2024 — Power Round — Problem 3

If { C } is a finite collection of closed sets, then U is a closed set. This is not true i i i i for arbitrary intersections. n The last important property is that of compactness: a set K ⊂ R is compact if and only if it is both closed and bounded; that is, its complement is open, and there exists some B ( r ) such that K ⊂ B ( r ). (0 ,..., 0) (0 ,..., 0) Theorem 1.0.1. Let K ⊃ K ⊃ K ⊃ ... 0 1 2 n be a countably infinite sequence of compact sets in R , each one of which containing the T rest of them. Suppose that they are all non-empty. Then K is also non-empty. i i Note that ∩ K is also compact since it is bounded and closed, being an intersection of i i closed sets. 2 Measures A measure is any way which we use to describe ”size” to sets, especially when seen as subsets of a larger space. They are designed to generalize and formalize the intuitive notions of length, area, volume, and probability that we know and love. Let X be a set, and Σ ⊂ P ( X ) a subset of the power set of X . Definition 2.0.1. Σ is called a σ -algebra on X if it is closed under complements, countable ∞ unions, and countable intersections. That is, if { E } ⊂ Σ ⊂ P ( X ), then i i =1

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