# Lower semicontinuous functions

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

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

is open in ${\displaystyle X}$ for all ${\displaystyle a\in \mathbb {R} }$.[1]

## Related Properties

• If${\displaystyle f}$ is lower semicontinuous and ${\displaystyle c\in [0,+\infty )}$ the ${\displaystyle cf}$ is lower semicontinuous.[2]
• If ${\displaystyle X}$ is a topological space and ${\displaystyle U\subset X}$ is any open set, then ${\displaystyle 1_{U}}$ is lower semicontinuous.[2]
• If ${\displaystyle f_{1},f_{2}}$ are lower semicontinuous, then ${\displaystyle f_{1}+f_{2}}$ is lower semicontinuous.[2]
• If ${\displaystyle X}$ is a locally compact Hausdorff space, and ${\displaystyle f:X\to \mathbb {R} \cup \{+\infty \}}$ is lower semicontinuous, then ${\displaystyle f(x)=\sup\{g(x):g\in C_{c}(X),0\leq g\leq f\}}$ where ${\displaystyle C_{c}(X)}$ denotes the space of all continuous functions on ${\displaystyle X}$ with compact support.[2]
• If ${\displaystyle \{x_{n}\}_{n\in \mathbb {N} }}$ is an convergent sequence in ${\displaystyle X}$ converging to some ${\displaystyle x_{0}}$, then ${\displaystyle f(x_{0})\leq \liminf _{n\to \infty }f(x_{n})}$.[1]
• If ${\displaystyle f:X\to \mathbb {R} \cup \{+\infty \}}$ is continuous, then it is lower semicontinuous. [1]
• In the case that ${\displaystyle X=\mathbb {R} }$, ${\displaystyle f}$ is Borel-measurable. [3]
• If ${\displaystyle {\mathcal {F}}}$ is a collection of lower semicontinuous functions from ${\displaystyle X}$ to ${\displaystyle \mathbb {R} \cup \{+\infty \}}$, then the function ${\displaystyle h(x):=\sup _{f\in {\mathcal {F}}}f(x)}$ is lower semicontinuous.[4]

## Lower Semicontinuous Envelope

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

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

## References

1. Craig, Katy. MATH 201A HW 1. UC Santa Barbara, Fall 2020.
2. Gerald B. Folland, Real Analysis: Modern Techniques and Their Applications, second edition, §7.2
3. Craig, Katy. MATH 201A HW 4. UC Santa Barbara, Fall 2020.
4. Craig, Katy. MATH 201A HW 5. UC Santa Barbara, Fall 2020.