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.
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