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In mathematics, a pre-measure is a set function that is, in some sense, a precursor to a bona fidemeasure on a given space. Indeed, one of the fundamental theorems in measure theory states that a pre-measure can be extended to a measure.
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Additionally, a semiring is a π-system where every complement is equal to a finite disjoint union of sets in
A semialgebra is a semiring that contains are arbitrary elements of and it is assumed that
It turns out that pre-measures give rise quite naturally to outer measures, which are defined for all subsets of the space More precisely, if is a pre-measure defined on a ring of subsets of the space then the set function defined by
is an outer measure on and the measure induced by on the -algebra of Carathéodory-measurable sets satisfies for (in particular, includes ). The infimum of the empty set is taken to be
(Note that there is some variation in the terminology used in the literature. For example, Rogers (1998) uses "measure" where this article uses the term "outer measure". Outer measures are not, in general, measures, since they may fail to be -additive.)
See also
Hahn-Kolmogorov theorem – Theorem extending pre-measures to measuresPages displaying short descriptions of redirect targets
References
Munroe, M. E. (1953). Introduction to measure and integration. Cambridge, Mass.: Addison-Wesley Publishing Company Inc. p. 310. MR0053186
Rogers, C. A. (1998). Hausdorff measures. Cambridge Mathematical Library (Third ed.). Cambridge: Cambridge University Press. p. 195. ISBN0-521-62491-6. MR1692618 (See section 1.2.)
Folland, G. B. (1999). Real Analysis. Pure and Applied Mathematics (Second ed.). New York: John Wiley & Sons, Inc. pp. 30–31. ISBN0-471-31716-0.