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

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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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2answers
36 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 ...
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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?
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0answers
18 views

A and B are independent under$\mathbb{P}$ but not under $\mathbb{Q}$

As the title, how to construct such an example that 2 events from the same measurable space ($\Omega$,$\mathscr{A}$) are independent under probability measure $\mathbb{P}$ but not independent under ...
3
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1answer
16 views

computing the Haar measure for O(n) and U(n) groups

My question is about how to compute the Haar measure for O(n) and U(n) groups. For example, for the conventional parametrization of SO(3) with 3 angels, the Haar measure is $ dO= ...
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4answers
361 views

What does it really mean when we say that the probability of something is zero?

Conventionally, people will say a probability of zero is equivalent as saying that the event is impossible. But when we look at the probability from a mathematics perspective, probability is defined ...
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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
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0answers
38 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?
4
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1answer
39 views

Can measure induce a topology on a Set?

When I was taught metric spaces in Topology, I came across the idea that metric defined on a set can induce a topology by creating a basis (open balls). If we have a measure defined on a set, can it ...
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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
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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 ...
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0answers
5 views

Robust Similarity Measure for template matching

I am now working on a specific pattern recognition(template matching) problem. Say I have a data bank with 5000 atom signals (entries in dictionary), each atom signal is an complex signal with length ...
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1answer
31 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 ...
3
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2answers
164 views

Is the Lebesgue measure of a set that is not dense = 0?

If I have a set that is not dense in any interval in $\mathbb{R}$, then can I say that it has zero Lebesgue measure? Or is there a counter example?
2
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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 ...
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1answer
28 views

The unit circle(disk), $\sigma(X)$ measurable function

If I have two measurable functions $X,Y:S \to \mathbb{R}$ (with the Lebesgue mesure) such that $X(\{x,y\})=x$ and $Y(\{x,y\})=y$ on the unit circle that is $x^2+y^2=1$. Then is $Y$ ...
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1answer
22 views

$\sigma$-algebras on a countable set is generated by partitions of the set

Let $X$ be a countable set, show that every $\sigma$-algebra is generated by a partition of $X$. I don't even know how to start. I 've been stuck on this problem for a long time. Any hints are ...
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1answer
30 views

Question about sigma algebra generated by a countable sequence of sets

I have a question about the following problem from the book "A course in Probability theory" (chapter 2, problem 9). Given a measurable space $(S,\Sigma)$, assume that $\Sigma$ is generated by $\{A_n ...
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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 ...
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1answer
14 views

The measures used to define Hausdorf dimension versus Haar measure

I don't know if there is a name for the measure, so let me construct it: Given a metric space $X$ we define the outer measures $H_\delta^\alpha$ by $\forall A \subset X$ $H_\delta^\alpha ...
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1answer
28 views

Hausdorf Dimension of a manifold of dimension n?

Let's say that $M$ is a differentiable manifold of dimension $n$. (This includes that $M$ is nonempty and second countable, so that it can be embedded into some Euclidean space, and is thus ...
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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 = ...
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0answers
11 views

Regard $C_{0}^{\infty}(U)$ as subspace of $L^{2}(U;m)$?

I have a quesiton about measure theory. Let $d \geq 3$, $U \subset \mathbb{R}^{d}$, open, and $dx$ be Lebesgue measure on $U$. Let $b_{i},d_{i},c \in L^{1}_{loc}(U;dx)$, $1\leq i \leq d$, such that ...
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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 ...
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1answer
26 views

Does a continuous function map a countable dense set to a countable dense set? [on hold]

Let $(\mathscr{X}_i, d_i), i =1,2$ be metric spaces. Let $f$ be a continuous function from $(\mathscr{X}_1,d_1)$ to $(\mathscr{X}_2,d_2)$. If $D \subset \mathscr{X}_1$ is countable and dense, is ...
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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 ...
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2answers
35 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, ...
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1answer
15 views

Soft: Interpretation of a periodic event on circle group

Recently I've been exploring probability measures on topological groups, derived from the (essentially) unique Haar measure defined thereon. I had begun to focus on the example of the circle group ...
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2answers
32 views

How to show that this set is Lebesgue measurable.

Let E be a Lebesgue measurable set. Show that $aE=\{a*x: x \in E\}$ is Lebegue measurable. My attempt: If a =0, then aE is either only the singleton consisting of 0, or the empty set. Both are ...
2
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1answer
26 views

Product $\sigma$ algebra and Borel sets

Consider $\mathbb{R}$ with the $\sigma$-algebra of Borel sets, and $\mathbb{R}^\mathbb{R}$ with the product $\sigma$-algebra(see p.22 of 'Real Analysis - Gerald B. Folland'). Does $[0,1]^\mathbb{R} ...
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1answer
13 views

outer measure and measurable sets

$\mu^*$ is an outer measure on $X$ and $\{A_j\}_{1}^\infty$ is a sequence of disjoint $\mu^*$-measurable sets. Prove that $\mu^*(E\cap(\bigcup_1^\infty A_j))=\sum_1^\infty \mu^*(E\cap A_j)$ for any ...
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0answers
16 views

Give me an example of an algebra that cannot be extended to a sigma algebra through finite/countable process of taking unions/intersections?

I've heard that finite or even countable process of taking unions/intersections can't make a collection of sets a sigma algebra. That is, if $\mathcal{A}\subseteq \mathcal{P}(M)$ is an algebra, and ...
3
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2answers
35 views

Product $\sigma$ algebra

Consider $\mathbb{R}$ with the $\sigma$-algebra of Borel sets, and $\mathbb{R}^\mathbb{R}$ with the product $\sigma$-algebra(see p.22 of 'Real Analysis - Gerald B. Folland'). Does $[0,1]^\mathbb{R} ...
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0answers
19 views

Outer measure on measure theory [on hold]

If $\mu^*$ is an outer measure on $X$ and $\{A_j\}_{1}^\infty$ is a sequence of disjoint $\mu^*$-measurable sets, then show that $\mu^*(E\cap(\bigcup_1^\infty A_j))=\sum_1^\infty \mu^*(E\cap A_j)$ for ...
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1answer
20 views

Measure Theory - finite measure spaces

Let $(X,\mathcal{M},\mu)$ be a finite measure space. Let $E,F \in\mathcal{M}$. Define $E \equiv F$ if $\mu(E\,\Delta\, F)=0$. Show that $\equiv$ is an equivalence relation on $\mathcal{M}$. I have ...
3
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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.
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1answer
36 views

(non)equivalence of definition of non-atomic measure for finitely additive measure

For a measure space $(X, \mathscr{B} ,\mu )$here're two definitions of non-atomic measure: Definition1: For each $A \in \mathscr{B}$, $\mu(A)> 0$, there exists $ B \subset A$ such that $0 < ...
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0answers
21 views

A finitely additive measure which is translation invariant and measure of interval is equal to length of interval?

Does there exist such a measure on all subsets of R: A finitely additive measure which is translation invariant and measure of interval is equal to length of interval
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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 ...
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2answers
27 views

Topology vs Borel sigma-algebra on a set X

What is the difference between: (X: a set) Topology (open set system) on X Borel sigma-algebra on X Both are a set of open subsets. Both include X and empty set. Both are Closed under union and ...
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1answer
25 views

Question about the construction of lebesgue measure of $(0,1]$ (or: the Borel sigma algebra is generated by the half open intervals)

I am reading the book "Probability with Martingales" In this book, the author constructs lebesgue measure on (0,1] as follows. Let $F =$ the collection of subsets of $(0,1]$ which can be written as ...
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1answer
19 views

Def. of total probability on wiki

As following: I am confused about "Bn is measurable". Bn is measurable means Bn is a sigmal-algebra. According to the definition of sigma-algebra: Bn must contain whole sample space, empty space; ...
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33 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 ...
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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 ...
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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 ...
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0answers
12 views

Measure Theory, Finite Positive Measure, Inf, Sup [on hold]

Starting off in Measure Theory, could use some help on this one: Let (X, M, μ) be a measurable space with finite positive measure and f : X → [−∞, ∞] a measurable function. (a) Show that for every ...
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0answers
10 views

Morphisms of $\mathsf{Meas}$ and Dynamical Systems

The morphisms of a category $\mathsf{Meas}$ whose objects are measure spaces are defined to be equivalence classes of a.e-equal measurable maps that pull back null sets to null sets. Why is pulling ...
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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
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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 ...
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1answer
24 views

What kind of f(n)'s make the limsup statement is true? What kind don't?

What kind of $f(n): \mathbb{N} \to \mathbb{N}$'s make the ff statement true? What kind don't? $\limsup A_{f(n)} \subseteq \limsup A_n$ where $n \in \mathbb{N}$ (*) Well obviously the answers to ...