Definition:Limit Inferior of Sequence of Sets/Definition 1
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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:
\(\ds \liminf_{n \mathop \to \infty} E_n\) | \(:=\) | \(\ds \bigcup_{n \mathop = 0}^\infty \bigcap_{i \mathop = n}^\infty E_n\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \paren {E_0 \cap E_1 \cap E_2 \cap \ldots} \cup \paren {E_1 \cap E_2 \cap E_3 \cap \ldots} \cup \cdots\) |
Also denoted as
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.
Also known as
The limit inferior of a sequence of sets is also known as its inferior limit.
Also see
Sources
- 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 $9$