0
votes
1answer
22 views

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 ...
5
votes
2answers
37 views

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 ...
0
votes
0answers
16 views

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?
0
votes
1answer
25 views

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 ...
4
votes
0answers
39 views

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?
0
votes
2answers
19 views

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 ...
3
votes
0answers
43 views

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 ...
1
vote
1answer
32 views

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 ...
2
votes
1answer
80 views

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 ...
0
votes
1answer
28 views

Existence of Monotone Sequence of Simple Functions

Let $\Omega$ be a measurable space with measurable sets $\Sigma$ and denote the space of simple functions by:$$\mathcal{S}:=\{s:\Omega\to\mathbb{R}:s=\sum_{k=1}^K ...
1
vote
0answers
25 views

Partial sums are alternate upper and lower bounds for $\mathbb{P}(\cup A_i)$

Show that $$ \sum_{k=1}^m(-1)^{k+1} S_k \leq \mathbb{P}(\cup_{i=1}^n A_i) \leq \sum_{k=1}^{m'}(-1)^{k+1} S_k$$ where $m, m' \leq n$, $m $ is even and $m'$ is odd, and $S_k = ...
0
votes
1answer
24 views

Measurability of Modulus

Context: This problem came up while reading an essay on Bochner integrability. Let $\Omega$ be a measure space and $E$ a Banach space. Consider two plain functions $f:\Omega\to E$ and $g:\Omega\to ...
1
vote
1answer
26 views

Can this exercise be solved by DCT, I was only able to use MCT.

How would you solve this exercise? You don't need to give me the details, just the general idea. Let f be a Lebesgue integrable function. Show that $\int f(x+a) d\lambda=f(x) d\lambda$ and ...
1
vote
2answers
36 views

Can the derivative of an absolutely continuous real function have a simple discontinuity?

If $f'$ exists everywhere, then we know that it cannot have any simple discontinuities. But in this case we only know that $f'$ exists a.e. (since $f$ is absolutely continuous). More specifically, ...
3
votes
0answers
38 views

Show that the set is a Borel set [duplicate]

$f:\mathbb R\to \mathbb R$ is a continuous function. Prove that the set of points $\{x \in \mathbb R |\ f$ is differentiable at $x \}$ is a Borel set. Any tips where to start? Thanks much.
0
votes
2answers
38 views

Additive but not $\sigma$-additive function

Give an example of a measure space $(\Omega, \mathit{F})$ and a function $\mu$ on $\mathit{F}$ that is additive but not $\sigma$-additive, i.e. $\mu(\cup A_i)= \sum\mu(A_i)$ for a finite collection of ...
1
vote
0answers
34 views

Prove that if $X \subset [a,b]$ isn't a measure-zero set, then there exists $\varepsilon >0$

Please, check my solution to this problem: "Prove that if $X \subset [a,b]$ isn't a measure-zero set, then there exists $\varepsilon >0$ such that, for every partition $P$ of $[a,b]$, the sum of ...
0
votes
0answers
13 views

Composition of $\mathcal B(\mathbb R)$-$\mathcal B(\mathbb R)$ and $\mathcal E$-$\mathcal B(\mathbb R)$-measurable functions.

I've proven $$h(x) = \left\{\begin{array}{ll} 1/x & : x \neq 0\\ 0 & : x = 0 \end{array} \right.$$ is $\mathcal B(\mathbb R)$-$\mathcal B(\mathbb R)$-measurable. I'm then asked to prove that ...
0
votes
0answers
22 views

Measure of triangular area

Let $\lambda\in[0,1]$, $\Omega=[0,1]^2$, $\vec{m}$ and $\vec{n}$ be two linearly independent vectors, $i\in\mathbb{N}$ and $h(t)$ the periodic extension of $$\tilde{h}(t):=\begin{cases} (1-\lambda)t ...
1
vote
2answers
14 views

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 ...
1
vote
1answer
21 views

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 ...
-2
votes
0answers
17 views

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 ...
0
votes
0answers
12 views

Lebesgue measure of decreasing sets. Something wrong with this proof?

$\mathcal{M}$ denotes the collection of Lebesgue-measurable sets and $\lambda$ the Lebesgue measure. I have the following exercise: Suppose $\lbrace E_n \rbrace^{\infty}_{n=1} \subset ...
1
vote
1answer
22 views

Is it necessarily true that the following intersection has a positive measure?

suppose that a set $S$ in $\mathbb{R}$ is a measurable set with measure greater than zero, and let $\mathcal{O}$ be an open cover of $S$, consisting of disjoint open intervals whose existence we know. ...
2
votes
1answer
45 views

$m^*(A \cup B) = m^*(A) + m^*(B)$

Let $A$ and $B$ be bounded sets for which there is an $\alpha >0$ such that for all $a \in A$ and all $b \in B$, $|a-b| \geq \alpha$. Prove that $m^*(A \cup B) = m^*(A) + m^*(B)$. I can't use ...
0
votes
0answers
47 views

$\sigma$-algebra that is countable, but not finite [duplicate]

I just had a chat with my real analysis teacher today, we talked about how there is no $\sigma$-algebra such that it has countably many elements, but finitely many. In the end, he said that I can ...
1
vote
1answer
24 views

show $[0,1]$ is uncountable using outer measure

This is a question from Real Analysis by Royden, 4th edition. (#5, pg. 34) Using properties of outer measure, prove that $[0,1]$ is uncountable. I believe that I am going to have to assume otherwise ...
0
votes
1answer
30 views

Measure of set of lines in $\mathbb{R}^2$

Suppose I have a set $A=\bigcup_{z_x} \bigcup_{z_y}A(z_x,z_y) $ where $z_x,z_y \in \mathbb{Z}$ and \begin{align} A(z_x,z_y)=\left\{ (x,y) \in \mathbb{R}^2: xz_x-yz_y=0 \right\} \end{align} The ...
0
votes
1answer
25 views

How to show $f(t, u(t))$ is measurable?

Given $f(\cdot, y)$ is measurable for each $y$, $f(x, \cdot)$ is continuous for each $x$. If $u(t)$ is continuous, how can I show that the function $f:[0,1]\times \mathbb{R} \rightarrow ...
1
vote
0answers
15 views

scalar dimension to the approximation of an integrable function

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 ...
1
vote
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. ...
0
votes
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 $X$ not empty. Define $$ l(A)=\begin{cases} \sup \left\{ \sum_{x \in F} f(x) \;\middle|\; ...
1
vote
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 ...
0
votes
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) ...
0
votes
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 ...
0
votes
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 ...
2
votes
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 ...
0
votes
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 ...
-3
votes
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.
0
votes
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.
1
vote
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 ...
3
votes
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 ...
2
votes
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 $m(E)< \infty$ ...
2
votes
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 ...
2
votes
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 ...
2
votes
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 ...
2
votes
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 ...
1
vote
1answer
24 views

Adapted and backward adapted?

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 $\mathcal{F}:=(\mathcal{F}_t)_{t\in [0,1]}$ ...
0
votes
2answers
46 views

Show that inverse image of a Lebesgue measurable function is Lebesgue-measurable

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 ...
1
vote
1answer
24 views

(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 ...