Content
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
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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) |
Content. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Content&oldid=17904