Let X a partition of U to k measurable sets : U = X _ 1 \ sqcup \ cdots \ sqcup X _ k.
42.
The ?-algebra of Borel sets is most popular, but not the only choice ( Baire sets, universally measurable sets etc . are used sometimes ).
43.
Given a countably additive measure ? on " X ", a measurable set is one that differs from a Borel set by a null set.
44.
The chief difference, many theologians say, is that Judaism and Islam embrace a far more exacting and measurable set of terms for repentance before offering forgiveness.
45.
A special case of the Lebesgue differentiation theorem is the Lebesgue density theorem, which is equivalent to the differentiation theorem for characteristic functions of measurable sets.
46.
One way of mathematically modeling a dissipative system is given in the article on " wandering sets " : it involves the action of a measurable set.
47.
One defines a family of measurable sets, which is very rich, and almost any set explicitly defined in most branches of mathematics will be among this family.
48.
Every probability measure is a continuous submeasure, so as the corresponding Boolean algebra of measurable sets modulo measure zero sets is complete it is a Maharam algebra.
49.
Non-measurable sets in a Euclidean space, on which the Lebesgue measure cannot be defined consistently, are necessarily complicated in the sense of being badly mixed up with their complement.
50.
The formulation of the definition of the generalized conic in the most general case when the cardinality of the focal set is infinite involves the notions of measurable sets and Lebesgue integration.
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