Document Type : Original Article

Authors

• Samwel Asamba ¹
• Benard Okelo ²
• Robert Obogi ³
• Priscah Omoke ²

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

Abstract

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.

Keywords

• Lower semi-continuous function
• Upper semi-continuous function
• $L^{p}$-space