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'limit superior of sets'
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| Title of object: |
limit superior of sets |
| Canonical Name: |
LimitSuperiorOfSets |
| Type: |
Definition |
| Created on: |
2002-12-08 07:59:11 |
| Modified on: |
2002-12-08 08:07:40 |
| Classification: |
msc:60A99, msc:28A05 |
| Defines: |
limit inferior of sets, infinitely often, i.o. |
Revision comment (for changes between this and next version):
| Changes for correction #4282 ('Object id is 3689, canonical name is LimitSuperiorOfSets'). |
Preamble:
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Content:
Let $A_1,A_2,\dots$ be a sequence of sets.
The limit superior of sets is defined by
\[\limsup A_n = \bigcap_{n=1}^\infty \bigcup_{k=n}^\infty A_k.\]
It is easy to see that $x\in \limsup A_n$ if and only if $x\in A_n$ for infinitely many values of $n$.
Because of this, in probability theory the notation $[A_n \operatorname{i.o.}]$ is often used to refer to $\limsup A_n$, where i.o. stands for \textit{infinitely often}.
The limit inferior of sets is defined by
\[\liminf A_n = \bigcup_{n=1}^\infty \bigcap_{k=n}^\infty A_k,\]
and it can be shown that $x\in A_n$ if and only if $x$ belongs to $A_n$ for all values of $n$ large enough. |
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