Lower semicontinuous functions

Let $X$ be a metric space (or more generally a topological space). A function $f:X\to \mathbb {R} \cup \{+\infty \}$ is lower semicontinuous if

$\{x\in X:f(x)>a\}=f^{-1}\left((a,+\infty ]\right)$

is open in $X$ for all $a\in \mathbb {R}$.

Related Properties

• If$f$ is lower semicontinuous and $c\in [0,+\infty )$ the $cf$ is lower semicontinuous.

• If $X$ is a topological space and $U\subset X$ is any open set, then $1_{U}$ is lower semicontinuous.

• If $f_{1},f_{2}$ are lower semicontinuous, then $f_{1}+f_{2}$ is lower semicontinuous.

• If $X$ is a locally compact Hausdorff space, and $f:X\to \mathbb {R} \cup \{+\infty \}$ is lower semicontinuous, then $f(x)=\sup\{g(x):g\in C_{c}(X),0\leq g\leq f\}$ where $C_{c}(X)$ denotes the space of all continuous functions on $X$ with compact support.

• If $\{x_{n}\}_{n\in \mathbb {N} }$ is an convergent sequence in $X$ converging to some $x_{0}$, then $f(x_{0})\leq \liminf _{n\to \infty }f(x_{n})$.

• If $f:X\to \mathbb {R} \cup \{+\infty \}$ is continuous, then it is lower semicontinuous.

• In the case that $X=\mathbb {R}$, $f$ is Borel-measurable.

• If ${\mathcal {F}}$ is a collection of lower semicontinuous functions from $X$ to $\mathbb {R} \cup \{+\infty \}$, then the function $h(x):=\sup _{f\in {\mathcal {F}}}f(x)$ is lower semicontinuous.

Lower Semicontinuous Envelope

Given any bounded function $f:X\to \mathbb {R}$, the lower semicontinuous envelope of $f$, denoted $f_{*}$ is the lower semicontinuous function defined as

$f_{*}(x)=\lim _{\epsilon \to 0}\inf\{f(y):d(x,y)<\epsilon \}=\inf\{\liminf _{n\to \infty }f(x_{n}):x_{n}\to x\}.$