A net of lower semicontinuous functions

Question

Assume we have a non-decreasing net of lower semicontinuous functions $f_\alpha:[0,1]\to\mathbb{R}$ such that $\lim_\alpha f_\alpha\to f$ pointwise. Please is it true that one can extract a countable sequence $\{f_n\}\subset \{f_\alpha\}$ such that $f_n\to f$ ? Any reference?

Answer 1

Take the subgraphs $$ U_\alpha = \{(x,y) \in [0,1]\times\mathbb R \mid y<f_\alpha(x)\} $$ which are open sets. The result in question would be a consequence of:

Let $X$ be a separable metric space. Let $\mathcal U = \{U_\alpha \mid \alpha\in A\}$ be a family of open subsets of $X$, directed by inclusion. Then there is a sequence $V_n \in \mathcal U, n \in \mathbb N$, also directed by inclusion, such that $\bigcup_\alpha U_\alpha=\bigcup_n V_n$.

Proof. The open set $\bigcup_\alpha U_\alpha$ is a separable metric space, hence second-countable, hence it has the Lindelof property (every open cover has a countable subcover). This gives us a sequence from $\mathcal U$ with union $\bigcup_\alpha U_\alpha$. Then use the directed property of $\mathcal U$ to get a directed sequnce with the same union.

Answer 2

Let us denote $I=[0,1]$ and let us choose $\varepsilon_n=2^{-n}$.

Notice that in the situation in the question we have $f=\sup\limits_{\alpha\in A} f_\alpha$, and thus $f$ is a lsc function.$\newcommand{\limti}[1]{\lim\limits_{#1\to\infty}}$

Theorem of Baire:¹ If $f$ is a lsc function, then there exists a monotone non-decreasing sequence $g_n$ of continuous functions such that $f=\limti n g_n$.

Using a sequence of continuous function described above, we get that for each $\alpha\in A$ and each $n\in\mathbb N$ the set $$M_{\alpha,n}=\{x\in I; f_\alpha(x)>g_n(x)-\varepsilon_n\}$$ is an open set. Moreover, for a fixed $n$ we see that these sets form an open cover of $I$, i.e., $$\bigcup_{\alpha\in A} M_{\alpha,n}=I.$$

From compactness we get that there is an open subcover, i.e., we have a finite set $F_n$ such that $\bigcup\limits_{\alpha\in F_n} M_{\alpha,n}=I$.

We choose $\alpha_n\in A$ such that such that $\alpha_n\ge \alpha$ for every $\alpha\in F_n$ (using the fact that $A$ is a directed set).

Now we have a sequence $\alpha_n$ such that for each $x\in I$ $$f(x)\ge f_{\alpha_n}(x) > g_n(x)-\varepsilon_n.$$ This already implies that for each $x\in I$ we have $$\limti n f_{\alpha_n}(x)=f(x),$$ i.e., we get pointwise convergence $f_{\alpha_n}\to f$. $\hspace{2cm}\square$

The use of compactness in combination with the lower semicontinuity is a bit similar to the proof of Dini's theorem for nets.²

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¹For example, Theorem 10.6 in A. C. M. van Rooij, W. H. Schikhof: A Second Course on Real Functions. This result is mentioned also in the Wikipedia article Semi-continuity (current revision, the Wikipedia article included Engelking as a reference).

²This result - formulated for nets of upper semicontinuous functions - can be found, for example, as Lemma 6.1.3 in Pedersen's Analysis Now, or as 17.7.j in Schechter's Handbook of Analysis and Its Foundations.