Lower Semicontinuous Function = Supremum of Sequence of Continuous Functions

metric-spacesreal-analysissemicontinuous-functions

Background

I'm reading Cedric Villani's Optimal Transport: Old and New [1] and came across a result (below) I'm not quite sure how to prove. It is used to prove Lemma 4.3 and through my research, I've found it to be known as "Baire's Theorem for Lower Semi-continuous Functions" with topological approaches found in other StackExchange posts like [3] and [4] but never formally worked out.

Question

If $(X, d)$ is a metric space and $F$ is a nonnegative lower semi-continuous function on $X$, then it can be written as the supremum of an increasing sequence of (uniformly?) continuous nonnegative functions. To see this, choose
$$
F_{n}(x) = \inf\limits_{y~\in~X}\{~ F(y) + n\cdot d(x,y) ~\}
$$
and show it is: (i) increasing; (ii) (uniformly?) continuous; (iii) convergent to $F$ [1, pg. 26; 2, pg. 55].

References:

  1. C. Villani, Optimal Transport, vol. 338. Berlin, Heidelberg: Springer Berlin Heidelberg, 2009. Available: https://ljk.imag.fr/membres/Emmanuel.Maitre/lib/exe/fetch.php?media=b07.stflour.pdf
  2. C. Villani, Topics in Optimal Transportation, 1st ed. American Mathematical Society, 2003.
  3. “Prove by definition that every upper semi-continuous function can be expressed as infimum of a sequence of continuous functions.,” Stack Exchange, 2017. [Online]. Available: https://math.stackexchange.com/questions/2227074/prove-by-definition-that-every-upper-semi-continuous-function-can-be-expressed-a?noredirect=1&lq=1. [Accessed: 28-Dec-2019].
  4. “Show that lower semicontinuous function is the supremum of an increasing sequence of continuous functions,” Stack Exchange, 2015. [Online]. Available: https://math.stackexchange.com/questions/1279763/show-that-lower-semicontinuous-function-is-the-supremum-of-an-increasing-sequenc/1284586. [Accessed: 28-Dec-2019].
  5. “What's behind the function g(x)=inf{f(p)+d(x,p):p∈X}?,” Stack Exchange, 2013. [Online]. Available: https://math.stackexchange.com/questions/616071/whats-behind-the-function-gx-operatornameinf-fpdx-pp-in-x?rq=1. [Accessed: 28-Dec-2019]

Best Answer

Hints:

$1).\ F(y) + n\cdot d(x,y)\le F(y) + (n+1)\cdot d(x,y)\Rightarrow F_n(x)\le F_{n+1}(x).$

$2).$ Since $F\ge 0$ we have by the triangle inequality $F(y)+nd(y,z)\le F(y)+nd(w,y)+nd(w,z)$ so $F_n(z)-F_n(w)\le nd(w,z)$. Now interchange $z$ and $w.$

$3).\ $ Fix $x_0\in X,\ \epsilon>0.$ Lower semicontinuity of $F$ implies that there is $\delta>0$ such that $d(x_0,y)<\delta\Rightarrow F(y)>F(x_0)-\epsilon.$ Now suppose $d(y,x_0)>\delta>0.$ Then, because $F\ge 0,$ there is a positive integer $n$ such that $F(y)+nd(x_0,y)>F(x_0).$ Now take the inf$:=\alpha$, over all such $y.$ Then, $\alpha\ge F(x_0)$. This shows that the inf in the definition of $F_n$ must lie in $B_{\delta}(x_0)$. But $y\in B_{\delta}(x_0)\Rightarrow F(y)+nd(x_0,y)\ge F(y)>F(x_0)-\epsilon\Rightarrow F_n(x_0)\ge F(x_0)-\epsilon.$

Remark: the proof goes through virtually unchanged if $F$ is only bounded below.

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