# Tagged Questions

Questions related to measures, sigma-algebras, measure spaces, Lebesgue integration and the like.

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### Show $A$ is an algebra of sets of $X$ [duplicate]

I'm having trouble with this problem, and not sure where to start with. I'm not sure if this is related to Borel algebra since they have very similar construction. Could someone help me with the ...
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### Functions h and r: prove $||h||_p = sup_{||r||_q \le 1} \int_E{rh}$

Functions h and r: prove $||h||_p = sup_{||r||_q \le 1} \int_E{rh}$, where $1/p + 1/q =1$. Using Holder's Inequality, I can prove that $||h||_p \le sup_{||r||_q \le 1} \int_E{rh}$, but I'm having ...
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### Can limit be taken inside the measure?

If $\lim_{n\rightarrow \infty }m(E_n) = 0$ is given, then can we take the limit inside the function? For example, in this case can I conclude $m(\lim_{n\rightarrow \infty } E_n) = 0$?
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### Existence of a locally essentially unbounded integrable function

Does there exist an integrable function $f\colon [0,1]\to \mathbb{R}_+$ such that for every $0\leq a < b\leq 1$ we have $\| \chi_{(a,b)} f\|_\infty = + \infty$?
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### Almost everywhere convergence of random variables

This is a question that my teacher is having us do for homework but I think there might be a typo in it. I was hoping if someone clear this up for me. The sequence $\{X_n\}$ of random variables ...
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### A Property of Martingale of Sum of i.i.d. Random Variables

I am trying to solve the following problem: Let $\{Y_n\}_{n=1}^{\infty}$ be a sequence of i.i.d. random variables with finite mean. Let $F_n =\sigma(Y_1,...,Y_n)$. Let $\tau$ be a stopping time ...
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### What are the two levels of probability theory?

What I mean is what is the Probability theory of using integrals called? (typically undergraduate course, Probability Theory I) Then what is the probability using measure theory? (typically graduate ...
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### Measurability of a set constructed from another measurable set

Let $E$ be a measurable subset of the real numbers. I define $\sigma(E) = \{ (x,y) \in \mathbb{R}^2 | x-y \in E\}$. I would like to prove that this set $\sigma(E)$ is Lebesgue measurable in ...
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### Measurability of integrals with respect to different measures [closed]

Let $Y$ be a locally compact Hausdorff topological space (further assumptions like metrizability, separability, etc., may be added if necessary) and let $\mathscr Y$ denote the Borel $\sigma$-algebra ...
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### What is the advantage of Borel sigma algebras in defining probability spaces?

I'm trying to get the central concepts correct, so I'm going to express them without embellishment. A Borel $\sigma$ algebra is defined as a sigma algebra generated by a topological space ...
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### Measurable Functions over finite or countables sets

Let $X$ be a countable or finite set and $u$ a measure over $(X,\Sigma)$. Let $A=${$A_1,A_2,...$} be the set of atoms of $(X,\Sigma)$. Prove that if $f: X\to \mathbb R$ is a measurable function, then ...
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### Prove $b-a \le \sum^n_{i=1}(b_i-a_i)$ by induction

Show that if the closed interval $[a,b]$ is covered by finitely many open intervals $(a_1,b_1), ...,(a_n,b_n)$, then $$b-a \le \sum^n_{i=1}(b_i-a_i)$$. I know that $(a_1,b_1), ...,(a_n,b_n)$ form an ...
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### Prove that half-open sets in $\mathbb{R}$ are measurable

Self-learning these concepts, so please be tolerant with imprecise terminology... Defining the standard topology on the real line $\mathbb {R}$ as all the open intervals, a Borel $\sigma$-algebra is ...
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### Lebesgue Integrability of $\left(\frac{1}{x}\right) \sin\left(\frac{1}{x}\right)$

Given $f:\mathbb{R}\rightarrow \mathbb{R}$ defined by $f(0)=0$ and $f(x)=\left(\frac{1}{x}\right) \sin\left(\frac{1}{x}\right)$ for $x\in \mathbb{R}-\{0\}$, can someone please give me a rigorous proof ...
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### How to recover a measure from the product

Let $(X,\mathcal{F}_X), (Y,\mathcal{F}_Y)$ be measurable spaces and $\mu :\mathcal{F}\rightarrow [0,\infty]$ be a measure (assume that $\mathcal{F}\supseteq \mathcal{F}_X\otimes\mathcal{F}_Y$). I do ...
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### I cast doubt on the theorem about function's measurability of semi-continuous function.

Def) $f$ is measurable if, for all $a$ in $\mathbb{R^1}$, $$\left\{\mathbf{x}:f(\mathbf{x})\gt a\right\} \text{ is measurable}~.$$ Def) $f$ is said to be upper-semi continuous if ...
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### Decreasing sequence of non-negative Lebesgue measurable functions and MCT

I'm learning about measure theory, specifically the Lebesgue integral of nonnegative functions, and need help with the following problem: Suppose that $f$ and $f_n$ are nonnegative measurable ...
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### What is the limit of the lebesgue integral of the function sequence fn=1/n

If $f_n=1/n$ then what is the value of the following limit (Lebesgue integral): $$\lim_{n\to\infty}\int_Rf_n$$ I basically want to prove that a generalisation of the monotone convergence theorem ...
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### Relation between Karamata's and Hardy-Littlewood's inequalities

In the field of (elementary) classical inequalities one of the most famous tools is the majorization inequality due to Karamata [1] (also known as Hardy-Littlewood-Polya). In its integral version, it ...
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### Poisson Process independent Wiener Process using singular measures

I was reading some stochastic calculus of Jump processes and saw the result that if $W_t$ is Brownian and $N_t$ is Poisson both adapted to the $W_t$'s natural filtration then these processes are ...
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### Characterization of the Jordan decomposition of a real-valued function of bounded variation from Folland.

This is a characterization of the Jordan decomposition of $F$ from Folland's Real Analysis. However, I can't see how the characterization makes sense. Let $F\in BV$ be a real valued function and ...
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### Proof Explaination: Show the set of measurable sets is closed under finite union

I have a proof of the above claim but I think there are some mistakes, I have highlighted them I hope someone could help figure out exactly what is wrong. Given $\omega$ an outer measure on set ...
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### What is the catch when introducing measure theory using $\sigma$-ring instead of $\sigma$-algebra?

I am currently using Matthew A Pons book Real Anaysis for Undergrad for introduction of measure theory In my opinion this book is unbelievably clear for almost all the sections EXCEPT the section ...
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### Calculating a probability of a maximum event

Let $\{X_n\}$ be a sequence of IID random variabels with continuous distribution function. For each n let $E_n = \{X_n>X_i \text{ for all } i <n \}$ be the event that there is a record at time ...
I am trying to understand the following argument given in a text book: Suppose $f \in L^1(\mathbb R^n)$, consider the function $\hat{f}(\zeta)= \int_{\mathbb R^n} \exp(-2\pi i X.\zeta)f(X)dX$. ...
Maximum Likelihood Estimation is quite clear to me when it is performed on finite sample sizes. The intuition of an obtained Maximum Likelihood estimate for given data $x_{1},...,x_{n}$, \$n \in ...