Characterization of semi-continuity in L^{p}-spaces

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