# Lower semi continuous envelope is lower semi continuous

Let X be a topological space and $$F:X \rightarrow \overline{\mathbb{R}}$$. The lower semi continuous envelope of $$F$$ is defined by $$sc^-F(u)=\sup\{\phi(u)\;|\; \phi :X \rightarrow \overline{\mathbb{R}} \; \mathrm{is \; l.s.c. \; and} \; \phi \leq F\}$$. Show that $$sc^-F$$ is lower semi continuous.

Here is my proof, but it seems a bit too easy, so it would be great if somebody could check if it is correct.

Let $$u_k \rightarrow u$$ in $$X$$. The set $$M:=\{\phi(u)\;|\; \phi :X \rightarrow \overline{\mathbb{R}} \; \mathrm{is \; l.s.c. \; and} \; \phi \leq F\}$$ is a family of l.s.c. functions, because every $$\phi \in M$$ is l.s.c. Then for every $$\phi \in M$$ we obtain: $$\phi(u) \leq \liminf \; \phi(u_k) \\ \leq \liminf \; \sup \{\phi(u_k)\;|\; \phi :X \rightarrow \overline{\mathbb{R}} \; \mathrm{is \; l.s.c. \; and} \; \phi \leq F\} = \liminf \; sc^-F(u_k).$$

Because this is true for every $$\phi \in M$$ it is also true for the supremum of all $$\phi$$ in $$M$$ (at this point I'm not really sure if it is correct) an we get

$$\sup\{\phi(u)\;|\; \phi :X \rightarrow \overline{\mathbb{R}} \; \mathrm{is \; l.s.c. \; and} \; \phi \leq F\} = sc^-F(u) \leq \liminf \; sc^-F(u_k)$$.

So the envelope of $$F$$ is l.s.c.

• If $X$ is not first-countable, it would be worth mentioning that $u_k$ is a net and how $\liminf$ is understood here. Otherwise, it looks ok to me. Alternatively, a proof via epigraphs is more straightforward (as intersection of closed sets is closed). – A.Γ. Dec 6 '18 at 11:03
• The set $M$ you define is not what you want it to be (it's a set of function values, not of functions). – MaoWao Dec 6 '18 at 11:15
• We haven't mentioned epigraphs in the lecture so I don't know how to prove this assumption this way. Thanks for the comments, I'll add it to the proof. – mathstu Dec 7 '18 at 9:20

Then for every $$\phi \in M$$ we obtain: $$\phi(u) \leq \liminf \; \phi(u_k) \\ \leq \liminf \; \sup \{\phi(u_k)\;|\; \phi :X \rightarrow \overline{\mathbb{R}} \; \mathrm{is \; l.s.c. \; and} \; \phi \leq F\} = \liminf \; sc^-F(u_k).$$
Here I strongly suggest to choose different variables for $$\phi$$ inside the set (or outside). For example:
$$\liminf \phi(u_k) \leq \liminf \; \sup \{\psi(u_k)\;|\; \psi :X \rightarrow \overline{\mathbb{R}} \; \mathrm{is \; l.s.c. \; and} \; \psi \leq F\} = \liminf \; sc^-F(u_k).$$