Proposition 5.22.2 ([Proposition 7.11, Fol99]).label Let $X$ be a topological space, then
- (1)
For any $U \subset X$ open, $\one_{U}$ is lower semicontinuous.
- (2)
For any $f: X \to (-\infty, \infty]$ lower semicontinuous and $\alpha \ge 0$, $\alpha f$ is lower semicontinuous.
- (3)
For any $f, g: X \to (-\infty, \infty]$ lower semicontinuous, $f + g$ is lower semicontinuous.
- (4)
For any collection $\mathcal{F}\subset (-\infty, \infty]^{X}$ of lower semicontinuous functions, $F = \sup_{f \in F}f$ is lower semicontinuous.
- (5)
For any $f: X \to (-\infty, \infty]$ lower semicontinuous, $f$ is Borel measurable.
Proof. (1): For any $\alpha \in \real$,
(2): If $\alpha = 0$, then $\alpha f = 0$ is continuous. If $\alpha > 0$, then for any $a \in \real$, $\bracs{\alpha f > a}= \bracs{f > a/\alpha}$ is open.
(3): Let $a \in \real$, $x_{0} \in \bracs{f + g > a}$, and $\eps \in (0, ((f + g)(x_{0}) - a)/2)$, then
As this holds for all $x_{0} \in \bracs{f + g > a}$, $\bracs{f + g > a}$ is open by Lemma 5.4.3.
(4): For any $a \in \real$,
is open.
(5): By Proposition 18.2.3.$\square$