Semicontinuous function
An extended real-valued function , defined on a complete metric space
, is said to be lower (upper) semi-continuous at a point
if
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The function is said to be lower (upper) semi-continuous on
if it is lower (upper) semi-continuous at all points
. The limit of a monotone increasing (decreasing) sequence of functions which are lower (upper) semi-continuous at a point
is again lower (upper) semi-continuous at
. If
and
are, respectively, lower and upper semi-continuous on
and for all
it is true that
,
,
, then there exists a continuous function
on
such that
for all
. If
is a non-negative regular Borel measure on
, then for any
-measurable function
there exist two monotone sequences of functions
and
satisfying the conditions: 1)
is lower semi-continuous,
is upper semi-continuous; 2) every
is bounded below, every
is bounded above; 3)
is a decreasing sequence and
is an increasing sequence; 4) for all
,
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5) -almost everywhere,
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and 6) if is
-summable over
,
, then
and
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(the Vitali–Carathéodory theorem).
References
[1] | I.P. Natanson, "Theory of functions of a real variable" , 1–2 , F. Ungar (1955–1961) (Translated from Russian) |
[2] | S. Saks, "Theory of the integral" , Hafner (1937) (Translated from French) |
Comments
The expressions lower semi-continuous and upper semi-continuous are often abbreviated to l.s.c. and u.s.c.. The notions of l.s.c. and u.s.c. function can be defined on any topological space . The superior (respectively, inferior) envelope of any family of continuous functions is l.s.c. (u.s.c.), and the converse is true whenever
is completely regular; this holds with a countable family of continuous functions if
is metrizable. Consequently, a semi-continuous function on a metric space
is of Baire class one (cf. Baire classes). The converse is not true.
Let . If
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then is of Baire class one, but neither lower nor upper semi-continuous. Also,
is lower semi-continuous, but
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Note that for all
, so
is the pointwise limit of an increasing sequence of continuous functions.
A very useful fact on semi-continuous functions is the Dini–Cartan lemma. Let be a compact space and
a family of l.s.c. functions such that for any finite subset
of
there is an
with
. If
is an u.s.c. function such that
, then there is an
such that
; in particular, one has
.
References
[a1] | K.R. Stromberg, "Introduction to classical real analysis" , Wadsworth (1981) |
Semicontinuous function. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Semicontinuous_function&oldid=27439