Namespaces
Variants
Actions

Content

From Encyclopedia of Mathematics
Revision as of 17:23, 7 February 2011 by 127.0.0.1 (talk) (Importing text file)
(diff) ← Older revision | Latest revision (diff) | Newer revision → (diff)
Jump to: navigation, search

A set has (Lebesgue) content zero if for all there is a finite set of closed rectangles such that and , where is Lebesgue measure.

More generally, let be a space equipped with a ring of subsets such that ( need not be a -ring and need not be in ). Let a function on be given such that for all , for at least one and such that is additive on . Such a function is called a content, and is the content of .

Define a rectangle as a product , where the are bounded closed, open or half-closed intervals, and let , where is the length of the interval . Define an elementary set in to be a finite union of rectangles. Let be the collection of all elementary sets. Each can be written as a finite disjoint union of rectangles ; then define . This defines a content on called Jordan content.

Given a content on and any , , one defines

where the infimum is taken over all finite sums such that , ; also one sets . This defines an outer measure on .

References

[a1] J.F. Randolph, "Basic real and abstract analysis" , Acad. Press (1968)
[a2] M.M. Rao, "Measure theory and integration" , Interscience (1987)
How to Cite This Entry:
Content. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Content&oldid=34243