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Riesz representation theorem, Borel measures and subsystems of second- order arithmetic. (English) Zbl 0770.03018

Summary: A formalized concept of finite Borel measures is developed in the language of second-order arithmetic. Formalization of the Riesz representation theorem is proved to be equivalent to arithmetical comprehension. Codes of Borel sets of complete separable metric spaces are defined and proved to be meaningful in the subsystem \(\text{ATR}_ 0\). Arithmetical transfinite recursion is enough to prove the measurability of Borel sets for any finite Borel measure on a compact complete separable metric space.

MSC:

03F35 Second- and higher-order arithmetic and fragments
28C15 Set functions and measures on topological spaces (regularity of measures, etc.)
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