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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Famous quotes containing the words outer and/or measure:
“The use of natural history is to give us aid in supernatural history: the use of the outer creation, to give us language for the beings and changes of the inward creation.”
—Ralph Waldo Emerson (1803–1882)
“Since we are assured that the all-wise Creator has observed the most exact proportions of number, weight and measure in the make of all things, the most likely way therefore to get any insight into the nature of those parts of the Creation which come within our observation must in all reason be to number, weigh and measure.”
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