Outer Measure - Outer Measure and Topology

Outer Measure and Topology

Suppose (X, d) is a metric space and φ an outer measure on X. If φ has the property that

whenever

then φ is called a metric outer measure.

Theorem. If φ is a metric outer measure on X, then every Borel subset of X is φ-measurable. (The Borel sets of X are the elements of the smallest σ-algebra generated by the open sets.)

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