Question
A partition of unity is “subordinate” to a collection of sets with this property if the support of each function of the partition is contained within the corresponding set with this property. For 10 points each:
[10e] Name this property that is possessed, by definition, by sets for which every point is an interior point.
ANSWER: open (The lead-in refers to open covers.)
[10h] A measure has the “outer” form of this property if the measure’s value on every Borel set is the infimum of the values the measure takes on open supersets.
ANSWER: regularity [accept outer regularity]
[10m] Given a positive linear functional, psi, this theorem gives a unique regular measure, mu, such that integrating any function f with respect to mu equals psi of f. Another theorem with this name gives an equivalence between operators and inner products on vector spaces.
ANSWER: Riesz representation theorem [accept Riesz–Markov–Kakutani representation theorem; prompt on partial answer] (Frigyes Riesz gave the initial statement of the Riesz–Markov–Kakutani representation theorem. Partitions of unity are used in one proof of the Riesz representation theorem.)
<Arya Karthik, Other Science - Mathematics>
Summary
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