Pre-measure

(Redirected from Pre-measures)

In mathematics, a pre-measure is a set function that is, in some sense, a precursor to a bona fide measure on a given space. Indeed, one of the fundamental theorems in measure theory states that a pre-measure can be extended to a measure.

Definition

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Let   be a ring of subsets (closed under union and relative complement) of a fixed set   and let   be a set function.   is called a pre-measure if   and, for every countable (or finite) sequence   of pairwise disjoint sets whose union lies in     The second property is called  -additivity.

Thus, what is missing for a pre-measure to be a measure is that it is not necessarily defined on a sigma-algebra (or a sigma-ring).

Carathéodory's extension theorem

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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

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  • Hahn-Kolmogorov theorem – Theorem extending pre-measures to measures

References

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  • 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. ISBN 0-521-62491-6. 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. ISBN 0-471-31716-0.