4.22 Semicontinuity
Definition 4.22.1 (Semicontinuous). Let $X$ be a topological space, $f: X \to (-\infty, \infty]$, and $g: X \to [-\infty, \infty)$, then $f$ is lower semicontinuous if for each $a \in \real$, $\bracs{f > \alpha}$ is open, and $g$ is upper semicontinuous if for each $a \in \real$, $\bracs{f < \alpha}$ is open.
Proposition 4.22.2. Let $X$ be a topological space, then
For any $U \subset X$ open, $\one_{U}$ is lower semicontinuous.
For any $f: X \to (-\infty, \infty]$ lower semicontinuous and $\alpha \ge 0$, $\alpha f$ is lower semicontinuous.
For any $f, g: X \to (-\infty, \infty]$ lower semicontinuous, $f + g$ is lower semicontinuous.
For any collection $\mathcal{F}\subset (-\infty, \infty]^{X}$ of lower semicontinuous functions, $F = \sup_{f \in F}f$ is lower semicontinuous.
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 4.4.3.
(4): For any $a \in \real$,
is open.
(5): By Proposition 14.2.3.$\square$
Proposition 4.22.3. Let $X$ be a LCH space and $f: X \to [0, \infty]$ be lower semicontinuous, then
Proof. Let $x \in X$ such that $f(x) > 0$ and $a \in (0, f(x))$, then $\bracs{f > a}$ is open. By Urysohn’s lemma, there exists $\phi \in C_{c}(X; [0, a])$ such that $\phi(x) = a$ and $\supp{\phi}\subset \bracs{f > a}$. As this holds for all $a \in (0, f(x))$,