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.)
Read more about this topic: Outer Measure
Famous quotes containing the words outer and/or measure:
“The outer world, from which we cower into our houses, seemed after all a gentle habitable place; and night after night a man’s bed, it seemed, was laid and waiting for him in the fields, where God keeps an open house.”
—Robert Louis Stevenson (1850–1894)
“This entire most beautiful order of good things is going to pass away after its measure has been exhausted; for both morning and evening were made in them.”
—St. Augustine (354–430)