Definition:Limit of Sets

Definition

Let $\Bbb S = \set {E_n : n \in \N}$ be a sequence of sets.

Let the limit superior of $\Bbb S$ be equal to the limit inferior of $\Bbb S$.

Then the limit of $\Bbb S$, denoted $\ds \lim_{n \mathop \to \infty} E_n$, is defined as:

$\ds \lim_{n \mathop \to \infty} E_n := \limsup_{n \mathop \to \infty} E_n$

and so also:

$\ds \lim_{n \mathop \to \infty} E_n := \liminf_{n \mathop \to \infty} E_n$

and $\Bbb S$ converges to the limit.

Also see

• Limit of Sets Exists iff Limit Inferior contains Limit Superior‎: all that is required for $\ds \lim_{n \mathop \to \infty} E_n$ to exist is for $\ds \limsup_{n \mathop \to \infty} E_n \subseteq \liminf_{n \mathop \to \infty} E_n$.

• Results about limits of sets can be found here.

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