22 Background Topology, open set, topological space, closed set Atopology on a set Xis a collectionT of subsets of X, calledopen sets, such that 1. ∅,X∈T, i.e. Acontains the empty set ∅ andXitself; 2. if {Uα ∈ T}α∈I is an arbitrary (finite or infinite) collection of open sets, then [α∈I Uα ∈T, i.e. their union is also open; 3. if {Ui ∈T} Ni=1 is a finite collection of open sets, then N\ i=1 Ui ∈T, i.e. their intersection is also open. The pair (X, T) is called a topological space, and the complement of an open set is called a closed set. Borel σ-algebra, Borel set Given a topological space(X, T), theBorel σ-algebra onXis the smallest σ-algebra that contains all open sets. A set that belongs to the Borel σ-algebra is called a Borel set. In other words, a Borel set is any set in a topological space that can be formed from open sets through the operations of countable union, countable intersection, and complement. Left translate, right translate Let Gbe a Lie group, then we define the left translate of a subset A⊆Gby an element g ∈Gas gA={g ◦a}a∈A, and the right translate as Ag ={a◦g}a∈A. Notably, left and right translates map Borel sets onto Borel sets.
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