# Tagged Questions

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### Measure of region

Let $\Omega:=[0,1]^2$, $f(x):=-x+1$ and $g(x):=(x-1)^2$. I am supposed to compute the $L^2$ measure of the area of the region given by $$M:=\{(x,y)\in\Omega\;|\;g(x)\leq y\leq f(x)\}.$$ Can I just ...
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### Prove Borel Measurable Set A with the following property has measure 0.

This question is exercise 4.10 of Richard F. Bass's Real Analysis for Graduate Students, 2nd edition. Let $\epsilon \in (0,1)$, let m be Lebesgue measure, and suppose A is Borel Measurable subset of ...
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### Sum of two measurable sets

I have heard that sum of two Lebesgue measurable sets in $\mathbb{R}$ may not be Lebesgue measurable. Can anyone give me an example with explanation?
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### complex measurable functions

I am trying to prove something about complex measurable functions. I have an idea for one direction and hope someone can give me a hint, I have gotten somee work done in this direction but need help ...
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### A Question From Measure Theory

How to show that a basis for the vector space $\mathbb{R}$ over the field $\mathbb{Q}$ is not Lebesgue measurable? Can anyone help me?
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### Describe the sigma algebra generated by singleton subsets

Let's denote the set of all singleton subsets of $X$(i.e. of all subsets consisting of one element) by $A$. Describe $\sigma(A)$ in the following two cases: i) $X$ is countable ii) $X$ is ...
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### Equivalence of Lebesgue Measurablity

Hello Mathematics Community. I am having some difficulties with the following problem dealing with Lebesgue Measure and its equivalent interpretation. I will first include the definitions which I am ...
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### Finding the limsup and liminf of a sequence of disks

Let $A_n$ be the interior of the circle with center at $( (-1)^n/n,0) )$ and radius $1$. In other words, $A_n$ = { $(x,y) | (x -(-1)^n/n )^2 + (y -0)^2 < 1$}. What is the $\limsup_n A_n$ and ...
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### Measure of intervals in the Borel sigma-algebra

This is an exercise from a real analysis book that is supposed to help you with entrance exams. I am trying to teach myself. Suppose $X$ is a set of real numbers, and $B$ is the Boresl ...
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### Let $f,g$ be $\mathcal E$-$\mathcal B(\mathbb R)$-measurable functions. I want to show piecewise function $h$ of $f$ and $g$ is also measurable.

Let $f,g$ be $\mathcal E$-$\mathcal B(\mathbb R)$-measurable functions. I want to show piecewise function $h$ of $f$ and $g$ is also measurable. Suppose $(X, \mathcal E)$ is a measure space, let ...
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### Let $X$ be a non-empty set, $A \subseteq X$. Decide the set $\mathcal M(\mathcal{E})$ of $\mathcal{E}$-$\mathcal B(\mathbb R)$-measureable functions.

Let $X$ be a non-empty set, $A \subseteq X$. Decide the set $\mathcal M(\mathcal{E})$ of $\mathcal{E}$-$\mathcal B(\mathbb R)$-measureable functions $f: X \rightarrow \mathbb R$ in each of the ...
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### Jordan Measurable Sets [on hold]

Let $\cdots \subset E_2 \subset E_1$ be a decreasing sequence of Jordan measurable sets in $R^n$ such that their intersection $Z = \bigcap_n^\infty E_i$ is a measure zero set. Prove that $Z$ is also ...
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Let $(M,\mathcal{A},\mu)$ space of probability. If $f\in L^1(\mu)$ then $f=\displaystyle{\lim_{n \rightarrow +\infty}\sum_{i=1}^n}\alpha_i\chi_{C_i}$ where $C_i\in \mathcal{A}$ and $\alpha_i \in ... 2answers 84 views ### How is every subset of real numbers measurable despite the existence of a non-measurable set? [closed] We know the existence of nonmeasurable subsets of$\mathbb R$by Vitali, but many books and lecture notes on real analysis still include the statement that every subset of real numbers is measurable. ... 1answer 38 views ### Showing that this is a measure Let$f:{X}\rightarrow\ [0, {\infty}]$a function and let$M=\mathcal{P}(X)$be a$\sigma$-algebra, with$Xnot empty. Define $$l(A)=\begin{cases} \sup \left\{ \sum_{x \in F} f(x) \;\middle|\; ... 1answer 49 views ### Property of the Riemann Integral Hello fellow Mathematics enthusiasts. I was hoping someone could help me with the following problem from Terry Tao's Introduction to Measure Theory: Let [a,b] be an interval, and let f,g:[a,b] \to ... 0answers 20 views ### Problem with measure theory argument. Could someone point out the flaw in this reasoning concerning the equality of outer and inner measure on sets with finite measure? Let \epsilon >0, and let S\subset \mathbb{R}^n with m^*(S) ... 1answer 24 views ### Divide a space into disjoint sets which has a small measure. triple (\Omega,\mathcal F,\mathbb P) ，\mathbb P is a finite measure. I have seen a statement in a textbook : "\forall \epsilon<\mathbb P(\Omega),we can divide \Omega into finite number of ... 1answer 19 views ### Question related to Measurable Function Question: Let f:\Omega \rightarrow [-\infty,\infty] be a measurable function. For any \epsilon > 0, there are a sequence of extended real numbers (a_k)_{k=1}^{\infty} and a sequence ... 1answer 15 views ### Dudley-Real Analysis and Probability(2e) -Lemma 3.1.8 - P90 A set F \subset X is called \mu^*-measurable, F \in M(\mu^*), iff for every set E \subset X, \mu^*(E)=\mu^*(E \cap F)+\mu^*(E\setminus F)$$3.1.8 Lemma M(\mu^*) is a \sigma-algebra ... 0answers 22 views ### Measure of a set A=\bigcup_{z_x,z_y} \{ (x,y) \in [0,1]^2 : |xAz_x-yBz_y| \le c \} What is the measure of the set: \begin{align} &A=\bigcup_{z_x,z_y} A(z_x,z_y)\\ &\text{ s.t. } 1 \le z_x \le N_x \text{ and } 1 \le z_y \le N_y \end{align} where A(z_x,z_y)=\{ (x,y) \in ... 0answers 20 views ### Show that the union of \sigma-algebra need not to be a \sigma-algebra? [duplicate] I want to show that the union of \sigma-algebra need not to be a \sigma-algebra. But I do not know how. 1answer 32 views ### Arbitrary union/intersection in \sigma-algebra It is well known that the definition of \sigma-algebra is concerned about countable operations.I want to find a \sigma-algebra that is not closed under arbitrary union/intersection. 0answers 31 views ### Proof monotone convergence theorem, why do they use this lim sup? I have a question about the proof of the MCT. First they use a lemma, this is ok, but I'll show it for completeness: Now comes the proof. But I am wondering, why do they use a lim sup here?, why ... 1answer 68 views ### Prove that \int_0^1f'(x)dx \leq f(1) - f(0). Let f(x) be a non-decreasing function on [0, 1]. You may assume that f is differentiable almost everywhere. Prove that \int_0^1f'(x)dx \leq f(1) - f(0). I am having a hard time with this ... 2answers 38 views ### How do we prove the following set is measurable? I was reading the proof of Egorov Theorem in the Real Analysis Book of Elias M Stein Suppose \{f_k\} is a sequence of measurable functions defined on the measurable set E with m(E)< \infty ... 0answers 31 views ### Borel measure and positive linear forms I'm just starting to learn about positive linear forms. If we call C_{C}(X) the space of all continuous functions with compact support from domain X and \mathbb{C} (with X a locally compact ... 0answers 23 views ### If X is a LCHS and f \in C_{C}(X) and \mu is a Borel measure, then f \in L^{1}(d\mu). I want to prove the following statement: If X is a locally compact Hausdorff topological space, and f \in C_{C}(X) (f is a continuous function with compact support), and if \mu is a Borel ... 1answer 95 views ### Equality about limsup. Suppose \sum_{n=1}^\infty \mathbb P(A_n)=\infty,then:\limsup_{n\to\infty}\frac{(\sum_{k=1}^n \mathbb P(A_k))^2}{\sum_{i,k=1}^n\mathbb P(A_i\cap A_k)}=\limsup_{n\to\infty}\frac{\sum_{1\le ... 1answer 65 views ### How to understand the exchangeable\sigma$-algebra? Suppose there are$(\Omega,\mathcal F,\mathbb P)$and r.v.$\xi_i$(i$\ge$1)$\xi_i:(\Omega,\mathcal F,\mathbb P)\to(\mathbb R,\mathcal B,\mu)A\in$the exchangeable$\sigma$-algebra$\mathcal E ...
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I understand the following: Consider a probability space $(\Omega, \mathcal{A},P)$ and a Brownian motion $B=\{B_t, t\in [0,1]\}$ on this space and denote $\mathcal{F}:=(\mathcal{F}_t)_{t\in [0,1]}$ ...
I am struggling with this exercise. Can anyone please give me a hint? Suppose f is Lebesgue-Measurable. Show that $f^{-1}(B)$ is Lebesgue- measurable for any borel set B. I do know that both ...
### (a) Prove that $f_n(x) → f(x)$ uniformly on $E$ as $n → ∞.$
Let $E ⊂ R$ be a compact (i.e., closed bounded) set of real numbers. Suppose $\{f_n\}$ is a sequence of real-valued continuous functions which converges pointwise on $E$ to a function $f$ that is also ...