Definition:Limit Superior of Sequence of Sets/Definition 2

Definition

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

• Equivalence of Definitions of Limit Superior of Sequence of Sets

Sources

• 1951: J.C. Burkill: The Lebesgue Integral ... (previous) ... (next): Chapter $\text {I}$: Sets of Points: $1 \cdot 1$. The algebra of sets
• 1970: Avner Friedman: Foundations of Modern Analysis ... (previous) ... (next): $\S 1.1$: Rings and Algebras