Lower semicontinuous functions: Difference between revisions

Latest revision as of 20:45, 10 December 2020

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 $a\in \mathbb {R}$ .^[1]

Related Properties

If $f$ is lower semicontinuous and $c\in [0,+\infty )$ the $cf$ is lower semicontinuous.^[2]

If $X$ is a topological space and $U\subset X$ is any open set, then $1_{U}$ is lower semicontinuous.^[2]

If $f_{1},f_{2}$ are lower semicontinuous, then $f_{1}+f_{2}$ is lower semicontinuous.^[2]

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.^[2]

If $\{x_{n}\}_{n\in \mathbb {N} }$ is an convergent sequence in $X$ converging to some $x_{0}$ , then $f(x_{0})\leq \liminf _{n\to \infty }f(x_{n})$ .^[1]

If $f:X\to \mathbb {R} \cup \{+\infty \}$ is continuous, then it is lower semicontinuous. ^[1]

In the case that $X=\mathbb {R}$ , $f$ is Borel-measurable. ^[3]

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.^[4]

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\}.

References

↑ ^1.0 ^1.1 ^1.2 Craig, Katy. MATH 201A HW 1. UC Santa Barbara, Fall 2020.
↑ ^2.0 ^2.1 ^2.2 ^2.3 Gerald B. Folland, Real Analysis: Modern Techniques and Their Applications, second edition, §7.2
↑ Craig, Katy. MATH 201A HW 4. UC Santa Barbara, Fall 2020.
↑ Craig, Katy. MATH 201A HW 5. UC Santa Barbara, Fall 2020.

[Craig-1] 1.0 ^1.1 ^1.2 Craig, Katy. MATH 201A HW 1. UC Santa Barbara, Fall 2020.

[Folland-2] 2.0 ^2.1 ^2.2 ^2.3 Gerald B. Folland, Real Analysis: Modern Techniques and Their Applications, second edition, §7.2

[Craig1-3] Craig, Katy. MATH 201A HW 4. UC Santa Barbara, Fall 2020.

[Craig2-4] Craig, Katy. MATH 201A HW 5. UC Santa Barbara, Fall 2020.

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[2]

[3]

[4]

@@ Line 1: / Line 1: @@
 Let <math> X </math> be a metric space (or more generally a topological space). A function <math> f : X \to \mathbb{R} \cup \{ +\infty \} </math> is '''lower semicontinuous''' if
 :<math> \{ x \in X : f(x) > a \} = f^{-1} \left( ( a , +\infty ] \right) </math>
-is open in <math> X </math> for all <math> a \in \mathbb{R} </math>.<ref name="Craig">Craig, Katy. ''MATH 201A HW 3''. UC Santa Barbara, Fall 2020.</ref>
+is open in <math> X </math> for all <math> a \in \mathbb{R} </math>.<ref name="Craig">Craig, Katy. ''MATH 201A HW 1''. UC Santa Barbara, Fall 2020.</ref>
+==Related Properties==
+*If<math> f</math> is lower semicontinuous and <math> c \in [0, +\infty) </math> the <math> cf</math> is lower semicontinuous.<ref name="Folland">Gerald B. Folland, ''Real Analysis: Modern Techniques and Their Applications, second edition'', §7.2 </ref>
+*If <math> X </math> is a topological space and <math> U \subset X </math> is any open set, then <math> 1_U </math> is lower semicontinuous.<ref name="Folland">Gerald B. Folland, ''Real Analysis: Modern Techniques and Their Applications, second edition'', §7.2 </ref>
+*If <math> f_1, f_2 </math> are lower semicontinuous, then <math> f_1+f_2 </math> is lower semicontinuous.<ref name="Folland">Gerald B. Folland, ''Real Analysis: Modern Techniques and Their Applications, second edition'', §7.2 </ref>
+*If <math> X </math> is a locally compact Hausdorff space, and <math> f:X \to \mathbb{R} \cup \{+\infty\} </math> is lower semicontinuous, then <math> f(x) = \sup \{g(x) : g \in C_c(X), 0 \leq g \leq f \} </math> where <math> C_c(X) </math> denotes the space of all continuous functions on <math> X </math> with compact support.<ref name="Folland">Gerald B. Folland, ''Real Analysis: Modern Techniques and Their Applications, second edition'', §7.2 </ref>
+*If <math> \{x_n\}_{n \in \mathbb{N}} </math> is an convergent sequence in <math>X </math> converging to some <math>x_0 </math>, then <math>f(x_0) \leq \liminf_{n \to \infty} f(x_n)  </math>.<ref name="Craig">Craig, Katy. ''MATH 201A HW 1''. UC Santa Barbara, Fall 2020.</ref>
+*If <math> f: X \to \mathbb{R} \cup \{+\infty\}</math> is continuous, then it is lower semicontinuous. <ref name="Craig">Craig, Katy. ''MATH 201A HW 1''. UC Santa Barbara, Fall 2020.</ref>
+*In the case that <math> X = \mathbb{R} </math>, <math> f </math> is Borel-measurable. <ref name="Craig1">Craig, Katy. ''MATH 201A HW 4''. UC Santa Barbara, Fall 2020.</ref>
+*If <math> \mathcal{F} </math> is a collection of lower semicontinuous functions from <math> X </math> to <math> \mathbb{R}\cup \{+\infty\} </math>, then the function <math> h(x) := \sup_{f \in \mathcal{F}} f(x) </math> is lower semicontinuous.<ref name="Craig2">Craig, Katy. ''MATH 201A HW 5''. UC Santa Barbara, Fall 2020.</ref>
+==Lower Semicontinuous Envelope==
+Given any bounded function <math> f :X \to \mathbb{R} </math>, the lower semicontinuous envelope of <math> f </math>, denoted <math> f_* </math> is the lower semicontinuous function defined as
+:<math> 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 \}. </math>
 ==References==

Lower semicontinuous functions: Difference between revisions

Latest revision as of 20:45, 10 December 2020

Related Properties

Lower Semicontinuous Envelope

References

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