Definition:Limit Inferior of Sequence of Sets/Definition 1

Definition

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

Then the limit inferior of $\sequence {E_n : n \in \N}$, denoted $\ds \liminf_{n \mathop \to \infty} E_n$, is defined as:

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

$:=$

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

$\ds $

$=$

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

The limit inferior of $E_n$ can also be seen denoted as:

$\ds \underset {n \mathop \to \infty} {\underline \lim} E_n$

but this notation is not used on $\mathsf{Pr} \infty \mathsf{fWiki}$.

Some sources merely present this as:

$\ds \underline \lim E_n$

The abbreviated notation $E_*$ can also be seen.

The limit inferior of a sequence of sets is also known as its inferior 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 $9$