the Limes superior to the sequence of sets . Alternative spellings are for the Limes inferior or
for the Limes superior.
example
Consider as an example the sequence of sets with
on the basic set . It is now
.
It follows directly from this
Analogously follows for the Limes superior
and thus
interpretation
The limes superior and inferior can be interpreted as follows:
You can make this clear from the formulas when you write out the outer set operation. It is then
Each of the quantities is written out
.
If one now combines all of the to form the Limes inferior, then the union contains all elements of the superset that are contained in at least one . This is equivalent to having an index for each element so that each contains if is. But this can only be the case if all but a finite number are contained in all .
The same results for the Limes superior
Then are the individual union sets
If you now cut all of them to form the limes superior, the intersection contains all that lie in each one . But then these are exactly the elements that lie in an infinite number .
Connection with characteristic functions
The characteristic functions of the limes inferior or limes superior of sets are the pointwise limes inferior or limes superior of the characteristic functions of the individual sets: Off
For
and
For
follows
analogously for lim sup.
Overall, then
and
.
use
The superior limit of set sequences is used in probability theory, for example, in the Borel-Cantelli lemma or in Kolmogorow's zero-one law , where they are typical examples of terminal events . More generally, limes superior and inferior are used to define convergence of set sequences . A sequence of sets converges when Limes inferior and superior coincide. This is the case, for example, if there is an index for each , so that either applies to all or to all . Convergent set sequences occur, for example, in measure theory .