On Set Convergence III

Where in this, third and final, post on the topic of set convergence we introduce the notion of Painleve-Kuratowski convergence of sequences of sets in topological spaces and we discuss about the unique properties of this notion of converges. We see that the K-limit of a constant sequence may not be equal to the elements of that sequence and many other interesting properties.

* Read first the previous posts: On Set Convergence I and II.

A sequence of sets \{C_n\}_{n} is said to converge in the Painlené-Kuratowski sense if \liminf_n C_n = \limsup_n C_n. This common limit, whenever it exists, will be denoted by K{-}\lim_n C_n or simply \lim_n C_n.

Since we already know that for any sequence \{C_n\}_{n} it is \liminf_n C_n \subseteq \limsup_n C_n, in order to show that \lim_n C_n exists and equals some set C, it suffices to show that:

\limsup_n C_n \subseteq C \subseteq \liminf_n C_n

As it follows from the closedness properties of the inner and the outer limit, the Kuratowski limit (whenever it exists) is a closed set. Later we will give conditions under which a closed set C satisfies an inclusion as in the inclusion above.

As a first example of a K-convergent sequence of sets we prove the following:

Proposition 11 (Sequences of balls). Let \{x_n\}_{n} be a sequence in \mathbb{R}^n such that x_n\to x and and \{\rho_n\}_{n}\subseteq\left[0,\infty \right) with \rho_n\to\rho<\infty. Then ,

ball-conv.png

Figure 1. Convergence of a sequence of balls as in Proposition 11.

\lim_{n}\mathcal{B}\left( x_n,\rho_n\right) = \mathrm{cl} \mathcal{B}\left( x,\rho\right)

Proof. It would be a waste of time to calculate the inner limit and the outer limit separately and then corroborate that the both equal the closed ball \mathrm{cl}\mathcal{B}(x,\rho). Instead, it suffices to show that the following inclusions hold:

\limsup_n \mathcal{B}(x_n,\rho_n) \overset{(2)}{\subseteq} K \overset{(1)}{\subseteq} \liminf_n \mathcal{B}(x_n,\rho_n)

where K=\mathrm{cl}\mathcal{B}(x,\rho).

(1). First let us note that the inner limit in this case is written as:

\liminf_n \mathcal{B}\left( x_n,\rho_n\right) = \left\{x \left|  {\begin{array}{c} \forall \varepsilon>0,\ \exists N\in\mathcal{N}_\infty,\\ \forall k\in N,\ z\in \mathcal{B}(x_k,\rho_k+\varepsilon) \end{array} } \right. \right\}

Assume that z\in\mathrm{cl}{\mathcal{B}(x,\rho)}, or what is the same that \left\| z-x\right\| \leq \rho.

For every \varepsilon>0 there is a N=N\left(\varepsilon\right)\in\mathbb{N} such that \left\| x_k - x \right\|<\tfrac{\varepsilon}{2} for all k\geq N.

This means that for all k\geq N, \left\| z-x_k \right\| \leq \left\|z-x \right\|+\left\|x_k-x \right\|<\rho+\tfrac{\varepsilon}{2}

But also \rho_k\to\rho, hence there is a M=M\left(\varepsilon\right)\in\mathbb{N} such that for all k\geq M, it is \left| \rho_k-\rho \right|<\tfrac{\varepsilon}{2}. So, for k\geq \max\left\{ N,M \right\} we have:

\left\| z-x_k \right\| < \rho_k+\varepsilon \Leftrightarrow z\in\mathcal{B}(x_k,\rho_k+\varepsilon)

Then, z\in\liminf_{n\to\infty}\mathcal{B}(x_n,\rho_n).

This way we have proved that \mathrm{cl}\mathcal{B}(x,\rho) \subseteq \liminf_n \mathcal{B}(x_n,\rho_n).

(2). The second step is to prove that the outer limit also converges to \mathrm{cl}{\mathcal{B}(x,\rho)}. For that we use the fact that

\limsup_n \mathcal{B}\left( x_n,\rho_n\right) =\left\{x \left|  {\begin{array}{c} \forall \varepsilon>0,\ \exists N\in\mathcal{N}_\infty^\#,\\ \forall k\in N,\ z\in \mathcal{B}(x_k,\rho_k+\varepsilon) \end{array} } \right. \right\}

This suggests that if z\in\limsup_n \mathcal{B}\left( x_n,\rho_n\right) then  for all \varepsilon>0, there exists a strictly increasing sequence of integers \{n_k\}_{k} such that z\in \mathcal{B}(x_{n_k},\rho_k+\varepsilon) for all k\in\mathbb{N}. It takes similar actions as in (1) to complete the proof. ♣

Under the same assumptions, the limit inferior of this sequence of open balls converges to an open ball; for example it is:

\mathrm{D{-}liminf}_{n\to\infty} \mathcal{B}(x_n,\rho_n) = \mathcal{B}(x,\rho)

A sequence of balls whose radii diverge to \infty converges (in the Painlevé-Kuratowski sense) to the whole space as stated in the following proposition:

Proposition 12 (Divergent balls). Let \{x_n\}_{n} be a sequence in \mathbb{R}^p such that x_n\to x and and \{\rho_n\}_{n}\subseteq\left[0,\infty \right) with \rho_n\to\infty. Then,

\lim_{n}\mathcal{B}\left( x_n,\rho_n\right) = \mathbb{R}^p

and

\lim_{n}\mathcal{B}\left( x_n,\rho_n\right)^c = \varnothing.

It is quite easy to verify the above example. A similar example refers to sequences of convex polytopes.

Proposition 13 (Convex hulls). Let \{x_n^i\}_{n} be a sequence of a set of points in a space \left(\mathcal{X},\tau\right) such that x_n^i\overset{n}{\to}x^i for \in \mathcal{I}. Then \mathrm{conv} \left\{x_n^i\right\}_i \overset{K}{\to}\mathrm{cl}\ \mathrm{conv}\left\{x^i\right\}.

Furthermore, limits of nested sequences of sets, either increasing or decreasing, are
particularly easy to calculate.

Proposition 14 (Nested sequences). Let \{C_n\}_{n} be a nested and increasing sequence of sets. Then it is convergent in the Painlevé-Kuratowski sense and \lim_n C_n = \mathrm{cl}\bigcup_{n\in\mathbb{N}} C_n.

Proof. Since C_0\subseteq C_1 \subseteq \ldots C_k \subseteq C_{k+1} \subseteq \ldots, if \{n_k\}_{k} is a cofinal subset of \mathbb{N}, then \bigcup_{k\in\mathbb{N}}C_{n_k}=\bigcup_{n\in\mathbb{N}}C_n and the equality
also holds for their closures. Therefore,

\liminf_n C_n = \bigcap_{\Sigma\in \mathcal{N}_\infty^\#}\mathrm{cl}\bigcup_{k\in\Sigma} C_k=\mathrm{cl}\bigcup_{k\in\mathbb{N}}C_k

Similarly, we carry out the calculation for the \limsup from which it follows that

\lim_n C_n = \mathrm{cl}\bigcup_{k\in\mathbb{N}}C_k

And this completes the proof. ♣

The Painlevé-Kuratowski convergence can be described using set inclusions which make it easy to check whether the K-limit of a given sequence of sets exists. Since for any sequence of sets it is \liminf_n C_n \subseteq \limsup_n C_n then it will be \lim_n C_n = C whenever:

\limsup_n C_n \subseteq C \subseteq \liminf_n C_n

It is therefore expedient to study under what conditions a given set is inside \liminf_n C_n or is a superset of \limsup_n C_n. Some first results are given in the following theorem:

Theorem 1. Let \{C_n\}_{n} be a sequence of sets in a Hausdorff topological space \left(\mathcal{X},\tau\right) and C be a closed set. Then it is C\subseteq \liminf_n C_n if and only if for every V\in\tau such that V\cap C \neq \varnothing, there exists a N\in\mathcal{N}_\infty such that C_n\cap V\neq \varnothing for all n\in N.

The following theorem provides conditions for the inclusion C\supseteq \liminf_n C_n to hold.

Theorem 2 (Hit-and-miss criteria). [1, Thm. 4.5] For C_n \subseteq X and C a closed set, (a) C\supseteq \limsup_n C_n if and only if for every compact set B\subset \subset \mathcal{X} with B\cap C=\varnothing, there exists N\in\mathcal{N}_\infty so that C_n\cap B=\varnothing for all n\in N. (b) C \subseteq \liminf_n C_n if and only if for every open set O\subseteq X, there is a cofinite set N \in \mathcal{N}_\infty such that C_n \cap O \neq \varnothing for all n \in N.

hit-and-miss.png

Figure 2. Illustration of the hit-and-miss criteria of Theorem 2.

In normed spaces, the above stated results can be restated using open balls instead of arbitrary open sets and closed balls instead of arbitrary compact sets. If the space is additionally first countable (every local topological basis has a countable sub-basis), then we can consider a countable collection of open sets (e.g. open balls).

The following theorem provides sufficient conditions for a sequence of sets to be K-convergent:

Theorem 3. Let \left( \mathcal{X}, \tau \right) be a Hausdorff topological space and \{C_n\}_{n} a sequence of subsets of \mathcal{X}. Let O\in\tau. If whenever the set N=\left\{ n|C_n\cap O\neq\varnothing \right\} is infinite, it is also cofinite, then \{C_n\}_{n} is K-convergent.

References

[1] R.T. Rockafellar and R.J-B. Wets, Variational Analysis, Grundlehren der mathematischen Wissenshaften, vol. 317, Springer, Dordrecht 2000, ISBN: 978-3-540-62772-2.
[2] G.Beer, Topologies on Closed and Convex Closed Sets, Mathematics and its applications, vol. 268, Kluwer Academic Publishers, Dordecht 1993, ISBN: 0-7923-2531-1.
[3] G. Beer, On Convergence of Closed Sets in a Metric Space and Distance Functions, Bulletin of the Australian Mathematical Society, vol. 31, 1985, pp. 421-432.

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