Tagged Questions
3
votes
1answer
49 views
Is it true that $\cup_{n=N}^{\infty}A_{n}\setminus\cap_{n=N}^{\infty}A_{n}=\cup_{n=N}^{\infty}A_{n}\triangle A_{n+1}$?
During an exam I have claimed that if $\{A_{n}\}_{n=1}^{\infty}$
then for any $N\in\mathbb{N}$
$$\limsup A_{n}\setminus\liminf ...
0
votes
1answer
620 views
limsup liminf of sequence of sets
Following up from the discussion here:
Liminf and Limsup of a sequence of sets
I wanted to confirm my understanding of these concepts with another example. Suppose we have: $a_n>0$, $b_n >1$ ...
1
vote
3answers
188 views
A problem in Sigma algebra.
How do I conceptualise this expression :
Let {$A_n$}$^{n=\infty}_{n=1}$ belong to sigma algebra $A$. Define, $\limsup\{A_{n}\}=\bigcap_{n=1}^{\infty}\{\bigcup_{m=n}^{\infty}A_{n}\}
$ and similarly ...
2
votes
3answers
2k views
lim sup and lim inf of sequence of sets.
I was wondering if someone would be so kind to provide a very simple explanation of lim sup and lim inf of s sequence of sets. For a sequence of subsets $A_n$ of a set $X$, the $\limsup A_n= ...
1
vote
3answers
611 views
Interpretation of {Infinitely Often} = {Almost Always}
I am trying to better understand what it means for a sequence $A_n$ of subsets of a set $S$
to be such that
$\bigcap^\infty_{n = 1} \bigcup^\infty_{m = n} A_n = \limsup A_n = \liminf A_n = ...
3
votes
2answers
814 views
liminf and limsup with characteristic (indicator) function
So first let me state my homework problem:
Let $X$ be a set, let $\{A_k\}$ be a sequence of subsets of $X$, let $B = \bigcup_{n=1}^{+\infty} \bigcap_{k=n}^{+\infty} A_k$, and let $C = ...
4
votes
1answer
804 views
limsup and liminf of a sequence of points in a set
My ways to define/write limsup and liminf of a sequence of points in a set $X$:
They come from what I have understood. If you have other ways of understanding, really appreciate if you can reply ...
6
votes
2answers
1k views
limsup and liminf of a sequence of subsets of a set
I am confused when reading Wikipedia's article on limsup and liminf of a sequence of subsets of a set $X$.
It says there are two different ways
to define them, but first gives what is common for ...
1
vote
2answers
538 views
Proof: Limit superior intersection
How to prove $\limsup(\{A_n \cup B_n\}) = \limsup(\{A_n\}) \cup \limsup(\{B_n\})$? Thanks!
0
votes
1answer
823 views
limit superior and limit inferior of the given sequence of sets
A sequence of sets is defined as $A_n=\{x \in [0,1] : |\sum_{i=0}^{n-1} 1_{[\frac{i}{2n},\frac{2i+1}{4n})} - 1_{[\frac{2i+1}{4n},\frac{i+1}{2n})}| \geq p\}$ for some positive $p\geq0$. What is ...
5
votes
3answers
2k views
limit inferior and superior for sets vs real numbers
I am looking for an intuitive explanation of $\liminf$ and $\limsup$ for sequence of sets and how it corresponds to $\liminf$ and $\limsup$ for sets of real numbers. I researched online but cannot ...
