Definition:Limit Superior of Sequence of Sets
Definition
Definition 1
Let $\sequence {E_n : n \in \N}$ be a sequence of sets.
Then the limit superior of $\sequence {E_n: n \in \N}$, denoted $\ds \limsup_{n \mathop \to \infty} E_n$, is defined as:
| \(\ds \limsup_{n \mathop \to \infty} E_n\) | \(:=\) | \(\ds \bigcap_{i \mathop = 0}^\infty \bigcup_{n \mathop = i}^\infty E_n\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds \paren {E_0 \cup E_1 \cup E_2 \cup \ldots} \cap \paren {E_1 \cup E_2 \cup E_3 \cup \ldots} \cap \ldots\) |
Definition 2
Let $\sequence {E_n: n \in \N}$ be a sequence of sets.
Then the limit superior of $\sequence {E_n: n \in \N}$, denoted $\ds \limsup_{n \mathop \to \infty} E_n$, is defined as:
- $\ds \limsup_{n \mathop \to \infty} E_n = \set {x : x \in E_i \text { for infinitely many } i}$
Also denoted as
The limit superior of $E_n$ can also be seen denoted as:
- $\ds \underset {n \mathop \to \infty} {\overline \lim} E_n$
but this notation is not used on $\mathsf{Pr} \infty \mathsf{fWiki}$.
Some sources merely present this as:
- $\ds \overline \lim E_n$
The abbreviated notation $E^*$ can also be seen.
Also known as
The limit superior of a sequence of sets is also known as its superior limit.
Also see
- Definition:Limit Inferior of Sequence of Sets
- Definition:Limit of Sets
- Definition:Infinitely Often (Probability Theory)
- Results about limits superior of set sequences can be found here.
Linguistic Note
The plural of limit superior is limits superior.
This is because limit is the noun and superior is the adjective qualifying that noun.