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

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 ,

5) -almost everywhere,

and 6) if is -summable over , , then and

(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

then is of Baire class one, but neither lower nor upper semi-continuous. Also, is lower semi-continuous, but

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)