# Tagged Questions

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### 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. ...
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### Probability measure, borel cantelli lemma

Let (En) be a sequence of events the probability space such that the probability of the sequence (En) goes to 0. I am trying to prove if the infinite sum of the probability of [the intersection of En ...
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### 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, ...
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### 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 ...
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### Lim sup of sequence of sets and theirs unions [closed]

I have to prove the following equality: Can somebody help me to prove this?
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### 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 ...
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### 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 $\limsup A_n=\bigcap_{k=1}^\infty\left(\bigcup_{n\geq k}A_n\right)$. ...
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### 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 $\lim_{b \to +\infty} \sup{X_b}$. ...
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### 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 $\limsup X_n$. ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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: $a_n>0$, $b_n >1$ ...
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### 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 ...
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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 $$... 0answers 99 views ### \limsup bounded almost everywhere Consider z \in \mathbb{R}^n and a sequence \{ z_i \}_{i=1}^{\infty} such that z_i \rightarrow z. Let \phi: \mathbb{R}^n \times X \rightarrow \mathbb{R}_{\geq 0}. X is unbounded. I wonder ... 1answer 831 views ### Tail sigma algebra and \limsup of a sequence of subsets On a set \Omega, there is a sequence of sigma algebras (\mathcal{F}_n)_{n \in \mathbb{N}}. The tail sigma algebra of (\mathcal{F}_n) is defined to be \cap_{n=1}^{\infty} ... 1answer 623 views ### Intuition behind \limsup and \liminf for probabilities I've come across these limits in Fatou's lemma, this got me massively confused. I'd be grateful if someone could explain the intuition behind limit suprema and limit infima of probabilities (or ... 2answers 711 views ### Do limits of sequences of sets come from a topology? In measure theory we frequently see the following definitions:$$\limsup_{n\to\infty} A_n = \bigcap_{n=1}^{\infty}\left(\bigcup_{j=n}^{\infty} A_j\right)\liminf_{n\to\infty} A_n = ...
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 ...
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 ...