inferior limit

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

liminfF=lim AFinfA. \lim\inf F = \lim_{A \in F} \inf A .

Similarly, the superior limit of FF is the limit of the suprema of the sets in FF:

limsupF=lim AFsupA. \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 FF to define a couple of nets in LL, each indexed by the sets in FF; as is usual with filters, we take ABA \leq B iff BAB \subseteq A. If LL is merely a poset and not a complete lattice, then we still use the same definitions, but they can only exist if infA\inf A or supA\sup A exists for sufficiently small AFA \in F.

If FF is merely a filter base, then precisely the same formulas give the inferior and superior limits of the filter generated by FF. 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

liminf nx n=lim ninf mnx m \lim\inf_{n \to \infty} x_n = \lim_{n \to \infty} \inf_{m \geq n} x_m


limsup nx n=lim nsup mnx m. \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(x_n)_n converges to a point x x_\infty iff there exist a monotone increasing net yy and a monotone decreasing net zz such that x =sup ny n=inf nz nx_\infty = \sup_n y_n = \inf_n z_n and, for each yy-index ii and zz-index jj, it is nn-eventually true that y ix nz jy_i \leq x_n \leq z_j. In this case, we can also write

liminfF=sup AFinfA \lim\inf F = \sup_{A \in F} \inf A


limsupF=inf AFsupA. \lim\sup F = \inf_{A \in F} \sup A .

The reason is that the net (infA) AF(\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 yy and a constant net for zz); similarly, (supA) AF(\sup A)_{A \in F} is monotone decreasing, with its limit the same as its infimum.

The symbols liminf\lim\inf and limsup\lim\sup come from the Latin ‘limes inferior’ and ‘limes superior’; saying ‘limit inferior’ in English because the symbol is ‘liminf\lim\inf’, while common, is like saying ‘logarithm natural’ because the symbol for the natural logarithm is ‘ln\ln’ (from the Latin ‘logarithmus naturalis’). Another variation is to read ‘liminf\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.