Regular set function
An additive function defined on a family of sets in a topological space whose total variation
(cf. Total variation of a function) satisfies the condition
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where denotes the interior of a set
and
the closure of a set
(and
,
,
are in the domain of definition of
). Every bounded additive regular set function, defined on a semi-ring of sets in a compact topological space, is countably additive (Aleksandrov's theorem).
The property of regularity can also be related to a measure, as a special case of a set function, and one speaks of a regular measure, defined on a topological space. For example, the Lebesgue measure is regular.
References
[1] | N. Dunford, J.T. Schwartz, "Linear operators. General theory" , 1 , Wiley (1988) |
[2] | A.D. Aleksandrov, "Additive set-functions in abstract spaces" Mat. Sb. , 9 (1941) pp. 563–628 (In Russian) |
Comments
Although a set function is called regular if it satisfies a property of approximation from below or above involving "nice" sets, the precise meaning of "regular" usually depends on the context (and on the author). For example, a (Carathéodory) outer measure is called regular if for every part
of
one has
, with
a
-measurable set containing
; if
is a topological space, the outer measure
is called Borel regular if Borel sets are
-measurable and if the
above can be taken Borel. On the other hand, if
is a metrizable space and
is a finite measure on the Borel
-field, then
is always regular in the sense of the article above. In this setting
is often called inner regular, or just regular, if for any Borel subset
one has
, with
a countable union of compact sets included in
, that is, if
is a Radon measure. Instead of calling
Radon, one nowadays most often says that it is tight.
Regular set function. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Regular_set_function&oldid=12357