Measure space
A measurable space with a measure
given on
(i.e. a countably-additive function
with values in
for which
; the latter property follows from additivity if the measure is finite, i.e. does not take the value
, or even if there is some
with
). The notation
is often shortened to
and one says that
is a measure on
; sometimes the notation is shortened to
. The basic case is when
is a
-algebra (cf. Algebra of sets) and
can be represented as
with
and
. In this case the measure is called (totally)
-finite (while if
, then it is called (totally) finite). Such is, e.g., the Lebesgue measure on
(cf. Lebesgue space). However, sometimes non-
-finite measures are encountered, such as, e.g., the
-dimensional Hausdorff measure on
for
. One may also encounter modifications in which
takes values in
, or complex or vector values, as well as cases when
is only finitely additive.
References
[1] | P.R. Halmos, "Measure theory" , v. Nostrand (1950) |
[2] | N. Dunford, J.T. Schwartz, "Linear operators. General theory" , 1 , Interscience (1958) |
Measure space. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Measure_space&oldid=14867