Graphical convergence

Where in this post we wonder whether pointwise convergence makes sense for sequences of set-valued mappings. We present some essential limitations of the pointwise limit and we motivate the notion of graphical convergence. What is more, we use animated GIFs to better illustrate and understand these new notions.

We say that a sequence of mappings F^\nu converges pointwise to a mapping F, denoted by F^\nu \overset{p}{\to} F, if for every x the sequence of sets \{F^\nu(x)\}_{\nu} converges (in the PK sense) to F(x). The pointwise limit (\mathrm{p{-}lim}_\nu F)(x) is exactly the limit of the sequence \{F^\nu(x)\}_{\nu} when this exists. We may naturally define (\mathrm{p{-}liminf}_\nu F^\nu)(x) to be the inner limit of \{F^\nu(x)\}_{\nu} and (\mathrm{p{-}limsup}_\nu F^\nu)(x) to be the corresponding outer limit.

In Figure 1 we show the convergence of a sequence of single-valued continuous functions to a discontinuous and neither lsc nor usc function.


Figure 1. Sequence of continuous single-valued functions which converge to a function which is neither lsc nor usc.

Using the definition of pointwise convergence, the convergence of a sequence of mappings is studied by the convergence of a sequence of sets of the form F^\nu \overset{p}{\to} F for every x separately. This definition, convenient as it may be, suffers from certain pathologies. For instance, in the single-valued case, it does not transfer lower/upper semicontinuity: the p-limit of a sequence of lsc (usc) functions may not be lsc (usc).

Most important, however, is the fact that the value \inf F^\nu (assuming it always exists), does not necessarily converge to the infimum of the limit, \inf \mathrm{p{-}lim}_\nu F^\nu. Similarly, the set of minimisers \arg\min_x F^\nu(x) (assuming it exists) is not guaranteed to converge to \arg\min_x \mathrm{p{-}lim}F(x).

Set-valued mappings can be fully identified by the graphs, therefore, sequences of set-valued mappings can be understood as sequences of their graphs \mathrm{gph}F^\nu, that is, sequences of sets. The set \limsup \mathrm{gph}F^\nu is the graph of a mapping which is called the graphical outer limit and it is denoted by \mathrm{g{-}limsup}_\nu F^\nu. It is

\mathrm{gph}(\mathrm{g{-}limsup}_\nu F^\nu) = \limsup_\nu (\mathrm{gph} F^\nu)

Likewise, the inner limit of the sequence \mathrm{gph}F^\nu is the graph of a mapping known as the graphical inner limit which is denoted by \mathrm{g{-}liminf}_\nu F^\nu. We then have

\mathrm{gph}(\mathrm{g{-}liminf}_\nu F^\nu) = \liminf_\nu (\mathrm{gph} F^\nu)

When the graphical inner and the graphical outer limit coincide, then, we say that the sequence is graphically convergent and the common limit is denoted by \mathrm{g{-}lim}_\nu F^\nu


Figure 2. The graphical and the pointwise limit of a sequence of functions.


Using the definitions of the inner and outer limits, we can see that

(\mathrm{g{-}limsup}_\nu F^\nu)(x) = \{y\mid \exists N\in \mathcal{N}_\infty^\#, x^\nu \overset{N}{\to} x, u^\nu \overset{N}{\to} u, u^\nu \in F^\nu(x^\nu) \}

where recall that \mathcal{N}_\infty^{\#} is the family of cofinal subsets of \mathbb{N}. Similarly, the graphical inner limit is the following function

(\mathrm{g{-}liminf}_\nu F^\nu)(x) = \{y\mid \exists N\in \mathcal{N}_\infty, x^\nu \overset{N}{\to} x, u^\nu \overset{N}{\to} u, u^\nu \in F^\nu(x^\nu) \}

The set-valued mappings \mathrm{g{-}liminf}_\nu F^\nu and \mathrm{g{-}limsup}_\nu F^\nu are always osc because their graphs are closed (as the PK limits of a sequence of sets).

The graphical limit of a single-valued function might be multi-valued. In particular, the graphical limit may exist but the function may not converge pointwise. We give a pertinent example in Figure 3.


Figure 3. Sequence of single-valued functions converges graphically to a set-valued mapping. The sequence does not converge pointwise.

We understand that the pointwise and the graphical limits of a sequence of function may be quite different one from the other. They may in fact coincide under additional assumptions, but, in general, we just know that

  1. \mathrm{p{-}liminf}_\nu F^\nu \subseteq \mathrm{g{-}liminf}_\nu F^\nu,
  2. \mathrm{p{-}liminf}_\nu F^\nu \subseteq \mathrm{p{-}limsup}_\nu F^\nu,
  3. \mathrm{g{-}liminf}_\nu F^\nu \subseteq \mathrm{g{-}limsup}_\nu F^\nu.

We know that \mathrm{p{-}lim}_\nu F^\nu \subseteq \mathrm{g{-}lim}_\nu F^\nu always holds, provided that both limits exist.


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