About the notion of limsup and liminf

Recently, I'm reading something about viscosity solution of a PDE, and the notions of limsup and liminf haunt me all the time. The following are some examples.

• The upper semi-continuous envelope of a function $z: \mathbb{R}^n \rightarrow \mathbb{R}$ is defined as $$ z^*(x) := \limsup_{x' \rightarrow x}\ z(x') $$

• Suppose for every partition $P$ of the interval $[0,T]$, we are given a (continuous) function $V^P: [0,T]\times \mathbb{R}^2 \rightarrow \mathbb{R}$. Then, we define $$ \bar{V}(t,x,y) := \limsup_{mesh(P)\rightarrow 0, (t',x',y')\rightarrow (t,x,y)} V^P(t',x',y') $$ where $mesh(P)$ is the mesh size of the partition.

• Suppose we have a sequence of functions $f_n: \mathbb{R}^n \rightarrow \mathbb{R}$, then we can define $$ g(x) := \limsup_{n\rightarrow \infty}\ \sup_{x'\in\mathbb{R}^n} f_n(x') $$

I have formally learned limsup and liminf only in terms of sequences, so I wonder

• How to interpret the above definitions of functions in precise mathematical language? ($\epsilon$-$\delta$ description will be great.)

• If there are two limiting processes like in the second example, is there any kind of order of computation?

• How to generalize the notions of limsup and liminf in more general settings?

If you are familiar with the convergence of nets you should not be surprised that limit superior and inferior can be defined for any net of real numbers.

For any net $(x_\alpha)_{\alpha\in I}$ of real numbers defined on a general directed set $(I,\preceq)$ limit superior of the net is defined as

$$\limsup x_\alpha = \lim_{\alpha\in I} \sup_{\beta\succeq\alpha} x_\beta=\inf_{\alpha\in I} \sup_{\beta\succeq\alpha} x_\beta.$$

If I remember correctly, it can be defined equivalently as the largest cluster point of the net.

Some references:

• Infinite Dimensional Analysis: A Hitchhiker's Guide By Charalambos D. Aliprantis, Kim C. Border, Page 32.

• An introduction to Banach space theory By Robert E. Megginson, Page 217. (And also some exercise at the end of that section.)

• Handbook of analysis and its foundations by Eric Schechter, Sections 7.43-7.47. (This books deals with a more general case of nets in complete lattices.)

• Topologies on closed and closed convex sets by Gerald Alan Beer, Page 2

• They are also mentioned at Wikipedia (current revision), but the definition given in the books I've mentioned is, in my opinion, better for anyone who learns about this notion for the first time.

I believe that all situations you've mentioned in your post can be considered as special cases of limsup and liminf of nets. But this approach will be probably useful only for someone who is familiar with using convergence of nets at least a little bit.

EDIT: Perhaps it's worth mentioning how exactly your situation can be transformed to the limsup and liminf of nets. I guess it suffices if I explain, what is the directed set to work with.

For $x'\to x$ in $\mathbb R^n$, the net could be $\mathbb R^n\setminus\{x\}$ directed by $x'\le y'$ $\Leftrightarrow$ $d(y',x)\le d(x',x)$. (You can choose any of equivalent metrics for $\mathbb R^n$.)

In the second case we could work with quadruples $(P,t',x',y')$ such that $P$ is a partition and $t'\ne t$, $x'\ne x$, $y'\ne y$. The ordering can be given by $(P_1,t'_1,x'_1,y'_1)\le (P_2,t'_2,x'_2,y'_2)$ $\Leftrightarrow$ $mesh(P_2)\le mesh(P_1)$ and $(t'_2,x'_2,y'_2)$ is closer to $(t,x,y)$ than $(t'_1,x'_1,y'_1)$. (Again, any of many equivalent metrics can be chosen.)

Note that these nets are very natural to choose - since convergence of this net is precisely the convergence you would work with, if you replaced $\limsup$ by $\lim$.

Regarding your example 2, let me simplify your setting a little bit and assume that one is given a real valued function $(P,x)\mapsto V(P,x)$ and a nonnegative function $P\mapsto m(P)$. Here $x$ is a real number and $P$ may belong to any set on which $m$ is defined, possibly with no additional structure (you may think of $m$ as your mesh).

How would you define the limit $L$ of $V(P,x)$ when $x\to x_0$, say, and $m(P)\to0$? I guess you would ask the following: $$ \forall\varepsilon\gt0, \exists\alpha\gt0, \exists \mu\gt0, \forall (x,P), [|x-x_0|\leqslant\alpha, m(P)\leqslant\mu ]\implies [|V(P,x)-L|\leqslant\varepsilon ]. $$ Likewise the limsup would be defined as $$ \color{red}{\limsup V(P,x)=\lim_{(\alpha,\mu)\to(0,0)}W(\alpha,\mu)}, $$ where, for every $(\alpha,\mu)$ such that $\alpha\gt0$ and $\mu\gt0$, $$ \color{purple}{W(\alpha,\mu)=\sup\mathcal V} \quad\text{where}\quad\color{purple}{\mathcal V=\{V(P,x);|x-x_0|\leqslant\alpha, m(P)\leqslant\mu\}}. $$ As in the usual case, the function $(\alpha,\mu)\mapsto W(\alpha,\mu)$ is nondecreasing hence the limit is also an infimum, that is, $$ \color{brown}{\lim_{(\alpha,\mu)\to(0,0)}W(\alpha,\mu)=\inf\mathcal W} \quad\text{where}\quad\color{brown}{\mathcal W=\{W(\alpha,\mu); \alpha>0,\mu>0\}}. $$