Document Type : Original Article


1 Department of Pure and Applied Mathematics, Kisii University, Box 408-40200, Kisii-Kenya

2 Department of Pure and Applied Mathematics, Jaramogi Oginga Odinga University of Science and Technology, Box 210-40601, Bondo, Kenya

3 Department of Pure and Applied Mathematics, Kisii University, Box 408-40200, Kisii, Kenya


Upper and lower semi-continuous functions are important in many areas and play a key role in optimization theory. This paper characterizes the lower and upper semi-continuity of $L^{p}$-space functions. We prove that a function $\vartheta:\mathcal L\rightarrow \overline{\mathbb R}$ is lower semi-continuous if and only if each convergent Moore-Smith sequence  $\{q_{j}\}_{j\in \mathbb N}$ converging to $q\in \mathcal L$ implies that $\int_{\mathcal L} \vartheta(q)d\mu\leq\liminf \int_{\mathcal L}\vartheta(q_{j})d\mu, \forall q\in \mathcal L$. We further show that the sum of any two proper lower semi-continuous functions is lower semi-continuous and the product of a lower semi-continuous function by a positive scalar gives a lower semi-continuous function and the case of upper semi-continuous functions follows analogously. Additionally, we prove that for a function in an $L^p$-space L if $\vartheta(\varphi)=\int_{\mathcal L}\varphi d\mu$ such that $\varphi$ is measurable with respect to a Borel measure $\mu$, then $\vartheta$ is upper semi-continuous.


[1] G. Beer, Upper semicontinuous functions and the Stone approximation theorem, J. Approx. Theory, 34 (1982), 1-11.
[2] Y. Chen, Y. Cho, and L. Yang Note on the Results with lower semicontinuity, Bull. Korean Math. Soc., 39(4) (2002), 535-541.
[3] E. Chong and S. Zak, An Introduction to Optimization, Fourth Edition, John Wiley and Sons, Inc., 2013.
[4] R. Correa and A. Hantoute, Lower semicontinuous convex relaxation in optimization, SIAM J. Optim., 23(1) (2013), 54-73.
[5] R. Fletcher, Practical Methods of Optimization, John Wiley and Sons, New York, 1986.
[6] F. Gool, Lower semicontinuous functions With values in a continuous lattice, Comment. Math. Univ. Carolina 33(3) (1992), 505-523.
[7] E. Hernandez and R. Lopez, A new notion of semi-continuity of vector functions and its properties, J. Optim., 69 (2020), 1831-1846.
[8] B. Jordan, Semicontinuous Functions and Convexity, University of Toronto, 2014.
[9] E. Kreyszig, Introductory Functional Analysis with Applications, John Wiley and Sons, 1978.
[10] P. Kumlin, A Note on Lp Spaces, Functional Analysis Lecture Notes, Chalmers, 2003.
[11] A. J. Kurdila and M. Zabarankin, Convex Functional Analysis, Systems and Control, Foundations and Applications, Birkhauser Verlag, Basel, 2005.
[12] A. Mirmostafaee, Points of upper and Lower semicontinuity of multivalued functions, Ukrain. Math. J., 69 (2017), 97-123.
[13] E. Montefusco, Lower semi-continuity of functionals via the concentration-compactness principle, J. Math. Anal. Appl. 263 (2001), 264-276.
[14] J. Moreau, Convexity and Duality in Functional Analysis and Optimization, Academic Press, New York, 1966.
[15] N. B. Okelo, On Certain Conditions for Convex Optimization in Hilbert Spaces, arxiv:1903.10177.Math.FA, 2019.
[16] S. Varagona, Inverse limits with upper semicontinuous bonding functions and Indecomposability, Houston J. Math., 37(3) (2011), 1017-1034
[17] J. P. Vial, Strong convexity of set and functions, J. Math. Econ., 9(1982), 187-205.
[18] Z. Wu, Uniform convergence theorems motivated by Dini’s theorem for a sequence of functions, J. Math. Anal., 11 (2020), 27-36.