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:
“I know, it must have been my imagination, but it makes me realize how desperately alone the Earth is. Hanging in space like a speck of food floating in the ocean. Sooner or later to be swallowed up by some creature floating by.... Time will tell, Dr. Mason. We can only wait and wonder. Wonder how, wonder when.”
—Tom Graeff. Young astronomer, Teenagers from Outer Space, after just seeing the invading spaceship through his telescope, and dismissing it (1959)
“Might was the measure of right.”
—Marcus Annaeus Lucan (39–65)