inferior limit

Let $L$ be a complete lattice with a convergence structure, and let $F$ be a filter on (the underlying set of) $L$. Then the **inferior limit** of $F$ is the limit of the infima of the sets in $F$ (if this limit exists):

$\lim\inf F = \lim_{A \in F} \inf A .$

Similarly, the **superior limit** of $F$ is the limit of the suprema of the sets in $F$:

$\lim\sup F = \lim_{A \in F} \sup A .$

(Of course, if the convergence structure is not Hausdorff, then there may be multiple inferior and superior limits.) Here we are taking $F$ to define a couple of nets in $L$, each indexed by the sets in $F$; as is usual with filters, we take $A \leq B$ iff $B \subseteq A$. If $L$ is merely a poset and not a complete lattice, then we still use the same definitions, but they can only exist if $\inf A$ or $\sup A$ exists for sufficiently small $A \in F$.

If $F$ is merely a filter base, then precisely the same formulas give the inferior and superior limits of the filter generated by $F$. Of course, we can also start with anything else that generates a filter in some way, such as a sequence or more generally a net. In this case, we can write

$\lim\inf_{n \to \infty} x_n = \lim_{n \to \infty} \inf_{m \geq n} x_m$

and

$\lim\sup_{n \to \infty} x_n = \lim_{n \to \infty} \sup_{m \geq n} x_m .$

Every partial order on a set defines a convergence structure (the order convergence?) under which a net $(x_n)_n$ converges to a point $x_\infty$ iff there exist a monotone increasing net $y$ and a monotone decreasing net $z$ such that $x_\infty = \sup_n y_n = \inf_n z_n$ and, for each $y$-index $i$ and $z$-index $j$, it is $n$-eventually true that $y_i \leq x_n \leq z_j$. In this case, we can also write

$\lim\inf F = \sup_{A \in F} \inf A$

and

$\lim\sup F = \inf_{A \in F} \sup A .$

The reason is that the net $(\inf A)_{A \in F}$ is monotone increasing, so that its limit in the order convergence is the same as its supremum (using itself for $y$ and a constant net for $z$); similarly, $(\sup A)_{A \in F}$ is monotone decreasing, with its limit the same as its infimum.

The symbols $\lim\inf$ and $\lim\sup$ come from the Latin ‘limes inferior’ and ‘limes superior’; saying ‘limit inferior’ in English because the symbol is ‘$\lim\inf$’, while common, is like saying ‘logarithm natural’ because the symbol for the natural logarithm is ‘$\ln$’ (from the Latin ‘logarithmus naturalis’). Another variation is to read ‘$\lim\inf$’ as ‘limit infimum’ and similarly for ‘limit supremum’, although this is etymologically incorrect. Sometimes one sees the more fully translated terms ‘lower limit’ and ‘upper limit’. On the other hand, in German, untranslated Latin is most common.

Last revised on February 5, 2019 at 13:38:34. See the history of this page for a list of all contributions to it.