Definition:Limit Superior of Sequence of Sets/Definition 1

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:

$\ds \limsup_{n \mathop \to \infty} E_n$

$:=$

$\ds \bigcap_{i \mathop = 0}^\infty \bigcup_{n \mathop = i}^\infty E_n$

$\ds $

$=$

$\ds \paren {E_0 \cup E_1 \cup E_2 \cup \ldots} \cap \paren {E_1 \cup E_2 \cup E_3 \cup \ldots} \cap \ldots$

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.

The limit superior of a sequence of sets is also known as its superior limit.

1951: J.C. Burkill: The Lebesgue Integral ... (previous) ... (next): Chapter $\text {I}$: Sets of Points: $1 \cdot 1$. The algebra of sets
2005: René L. Schilling: Measures, Integrals and Martingales ... (previous) ... (next): $\S 9$: Problem $9$