1
vote
1answer
17 views

Coin tossing, two heads always followed by two tails - lim sup necessary?

In Bernoulli Space $\Omega$, let $E_n$ be the event that the $n$th toss is heads. Write down a formula in terms of the $E_n$ for the following event: “Every time two Heads appear in succession, the ...
3
votes
1answer
47 views

Trying to prove lim inf $(A_n) \subseteq$ lim sup $(A_n)$

I am trying to show that for a sequence of sets $(A_n)$ $$ lim \ inf \ (A_n) \subseteq lim \ sup \ (A_n)$$ I can see why this is true intuitively as follows - $x \in lim \ inf \ (A_n)$ $\implies ...
1
vote
1answer
73 views

measure space problem.

Let $(\Omega,\mathcal{F},\mu) $ be a probability space. Let $\delta>0$ and for each $n\in \mathbb{N}$. Let $A_n \in \mathcal{F}$ satisfy $\mu(A_n)\ge\delta$. Prove that the set $A_\infty $ ...
2
votes
1answer
29 views

What is the difference between limit inferior and limit?

So, I am working on problems on $L_p$ spaces. Every time I think of taking limits of both sides of an equation, the solution seems to take limit inferior/ limit superior instead and use the relation ...
1
vote
0answers
42 views

Can I conclude the following about limsup

I am trying to show that if $F:[a,b]\rightarrow\mathbb{R}$ is continuous and of bounded variation then $g(x)=\limsup_{h\rightarrow 0, h>0} \frac{F(x+h)-F(x)}{h}$ is a Lebesgue measurable function. ...
1
vote
1answer
136 views

Probability of limsup of a sequence of sets, Borel-Cantelli lemma,

Let $(E_n)$ be a sequence of events in a probability space such that the probability of $(E_n)$ goes to $0$. I am trying to prove that if $$\sum_{n=1}^\infty P (E_n\setminus E_{n+1}) <\infty$$ ...
3
votes
1answer
120 views

Please check if my proof is correct of Monotone Convergent theorem

I was required to prove Monotone Convergent Theorem as a corollary of Fatou lemma,i.e using Fatou lemma to prove the MCT. The hint I was given is let $f_n$ be a sequence of increasing function, ...
0
votes
1answer
56 views

Prove that $\sup_k f_k, \inf_k f_k, \lim \sup_k f_k, \lim \inf_k f_k, \lim_k f_k$ (if it exists) are all M-measurable.

If $f_1, f_2, f_3,...$ are $M$-measurable, prove that $\sup_k f_k, \inf_k f_k, \lim \sup_k f_k, \lim \inf_k f_k, \lim_k f_k$ (if it exists) are all M-measurable. My thoughts: We know for any sequence ...
2
votes
2answers
136 views

Lim sup of sequence of sets and theirs unions [closed]

I have to prove the following equality: Can somebody help me to prove this?
0
votes
1answer
145 views

Understanding the supremum limit of a set

Given a sequence ${A_n}$, we define the set lim sup $A_n = \{x : x$ belongs to infinitely many $A_n$'s$\}$ That is - lim sup $A_n = \bigcap_{m=1}^\infty (\bigcup_{n=m}^\infty A_n)$ I can't see how ...
1
vote
1answer
269 views

Liminf and Limsup in measure theory and in sequences

In measure theory, given sets $A_1,A_2,\ldots$, we define $\liminf A_n=\bigcup_{k=1}^\infty\left(\bigcap_{n\geq k}A_n\right)$ and $\limsup A_n=\bigcap_{k=1}^\infty\left(\bigcup_{n\geq k}A_n\right)$. ...
0
votes
1answer
132 views

Find the limit sup and limit inf of a given sequence of sets

Suppose we have a set $X_b = \{\frac{a}{b}:a \in \mathbb{Z^{+}}\} $ where $b \in \mathbb{Z^{+}}$. We want to find $\lim_{b \to +\infty} \inf{X_b}$ and also find find $\lim_{b \to +\infty} \sup{X_b}$. ...
1
vote
2answers
78 views

Showing independence of a limsup of an independent sequence

Let $\{X_n\}_{n \geq 1}$ be an independent sequence of random variables on $(\Omega, \mathcal{F}, \mathbb{P})$. Fix $n \geq 1$. I want to prove that $X_1, \ldots, X_n$ is independent of $\limsup X_n$. ...
1
vote
2answers
295 views

Probability of limsup

Let $A_1, A_2, A_3, \dots$ be a sequence of independent events on $\left (\Omega, \mathbb A, \mathbb P\right )$ such that $\mathbb P(A_n) < 1$ and $\mathbb P\left ...
0
votes
1answer
157 views

How to prove the limsup equals liminf for a monotone class.

How to prove if a class is monotone, then its limit supremum equals its limit infimum. Example, ${A_{n}}$ is a monotone class with $A_{n} \subset \Omega$, and $A_{1} \subset A_{2} \subset A_{3}... $, ...
1
vote
0answers
66 views

Probability measures on $\mathbb{T}$ whose Fourier coefficients tend to 1

Let $\mu$ be a probability measure on the complex unit circle $\mathbb{T}$. Are the following two assertions equivalent? $\limsup_{n\to\infty}|\hat{\mu}(n)|=1$. There exists an increasing sequence ...
1
vote
3answers
236 views

Fatou's Lemma Strengthened to Equality

I'm trying to use an example to show that Fatou's lemma can not be strengthened to equality. I was given a hint, which I'm not quite sure how to use. I was told that if I look at the one-dimensional ...
4
votes
1answer
203 views

limsup of intersection of events as a subset of intersection of limsups

Let $A_1, A_2, \ldots$ and $B_1, B_2, \ldots$ be two sequences of events in some probability triple $(\Omega, \mathcal{F}, \mathbf{P})$. Now, it is true that $\left(\limsup_n A_n\right) \cap ...
0
votes
2answers
480 views

Limits of infimum and supremum for sequences of functions

I need to show that $-\infty \leq \liminf_{k \to \infty}f_k \leq \limsup_{k \to \infty}f_k \leq \infty$ , where $f_k$ is a sequence of functions from $\mathbb{R}^n$ to $\mathbb{R}$. This seems ...
3
votes
0answers
122 views

Limit superior and limit inferior [duplicate]

Possible Duplicate: liminf and limsup with characteristic (indicator) function Suppose $\{E_k\}_{k\geq 1}$ is a sequence of measurable sets. Then we can define ...
1
vote
1answer
292 views

Upper semi continuity of Lim sup

I have been asked to prove If $f$ is bounded, then $g(x)= \overline{\lim}_{y\to x} f(y)$ is upper semi continuous. This means somehow I have to show that for some $x_0$ $$\overline{\lim}_{x\to ...
5
votes
1answer
87 views

Inequality for Fourier transform of measure

I am having trouble with the following question. Let $\mu$ be finite measure on $\mathbb{R}$ and let $\hat{\mu}(\xi) = \int_{-\infty}^\infty e^{-ix \xi} d\mu(x)$ be its Fourier transform. Prove that ...
2
votes
1answer
690 views

Limit inferior/superior of sequence of sets

Let $(\Omega, \mathcal{A}, \mu)$ be a measure space, where $\mu(\Omega)< \infty$. Further $(A_n)_{n \in \mathbb{N}}$ is a a sequence of $\mathcal{A}$-measurable sets. I want to prove, that $$ \mu ...
0
votes
1answer
570 views

lim sup of sequence of continuous function from $[0,1]\rightarrow [0,1]$

$f_n:[0,1]\to [0,1]$ be a continuous function and let $f:[0,1]\to [0,1]$ be defined by $$f(x)=\operatorname{lim\;sup}\limits_{n\rightarrow\infty}\; f_n(x)$$ Then $f$ is continuous and ...
3
votes
1answer
120 views

Form of $\sigma(X_n)$ and $\sigma(X_n)$-measurability

Problem: Suppose $\tilde{X}=(X_1,X_2,...)$ is a sequence of RVs on $(\Omega,\mathcal{B})$. Prove that $\sigma(\tilde{X})$ is generated by events of the form: $\bigcap_{i=1}^m \{X_i\leq x_i\}$ for ...
0
votes
1answer
455 views

Limit Superior of Random Variables

Suppose $X_n$ are iid random variables with $\mathbb{P}(X_n\le x)=1-e^{-x}$. By using the Borel Cantelli Lemmas it's fairly easy to show that $\mathbb{P}(\lim\sup X_n/\log n=1)=1$. My lecture notes go ...
0
votes
1answer
864 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
268 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 ...
1
vote
1answer
311 views

Bounded $\limsup$ integral implies $\limsup$ bounded almost everywhere?

Consider $z \in \mathbb{R}^n$ and $\{ z_i \}_{i=1}^{\infty}$ with $z_i \rightarrow z$. Let $\phi: \mathbb{R}^n \times X \rightarrow \mathbb{R}_{\geq 0}$. $X$ is unbounded. I'm wondering if $$ ...
0
votes
0answers
107 views

$\limsup$ bounded almost everywhere

Consider $z \in \mathbb{R}^n$ and a sequence $\{ z_i \}_{i=1}^{\infty}$ such that $z_i \rightarrow z$. Let $\phi: \mathbb{R}^n \times X \rightarrow \mathbb{R}_{\geq 0}$. $X$ is unbounded. I wonder ...
2
votes
1answer
988 views

Tail sigma algebra and $\limsup$ of a sequence of subsets

On a set $\Omega$, there is a sequence of sigma algebras $(\mathcal{F}_n)_{n \in \mathbb{N}}$. The tail sigma algebra of $(\mathcal{F}_n)$ is defined to be $\cap_{n=1}^{\infty} ...
0
votes
1answer
720 views

Intuition behind $\limsup$ and $\liminf$ for probabilities

I've come across these limits in Fatou's lemma, this got me massively confused. I'd be grateful if someone could explain the intuition behind limit suprema and limit infima of probabilities (or ...
6
votes
2answers
737 views

Do limits of sequences of sets come from a topology?

In measure theory we frequently see the following definitions: $$\limsup_{n\to\infty} A_n = \bigcap_{n=1}^{\infty}\left(\bigcup_{j=n}^{\infty} A_j\right)$$ $$\liminf_{n\to\infty} A_n = ...
0
votes
1answer
994 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 ...
6
votes
3answers
3k 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 ...