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 heartless and enormous Outer Black ...”
—Robert Frost (1874–1963)
“I do seriously believe that if we can measure among the States the benefits resulting from the preservation of the Union, the rebellious States have the larger share. It destroyed an institution that was their destruction. It opened the way for a commercial life that, if they will only embrace it and face the light, means to them a development that shall rival the best attainments of the greatest of our States.”
—Benjamin Harrison (1833–1901)