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

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1answer
81 views

Does every Lebesgue measurable set $A$ with $m(A)>0$ contain at least an open subset?

Does every Lebesgue measurable set $A$ with $m(A)>0$ contain at least a non-empty open subset? I came across this question when I was reading the Stein and Shakarchi's Real Analysis book. Is ...
2
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0answers
68 views

Compute the limit of $\int_\mathbb{R} f(x)\sin (nx)$ when $n\to\infty$, for $f \in L^1$

Let $f \in L^1(\mathbb{R})$. Find $$ \lim_{n \rightarrow \infty} \int_{-\infty}^\infty f(x)\sin(nx) dx \,. $$ LDCT is a no go, as well as MCT and FL, which are really the only integration ...
0
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1answer
25 views

For a 2-dimensional random vector, express the probability of being in a rectangle in terms of CDF

For $a_1\le b_1$ and $a_2\le b_2$, show that $$P\{ a_1<X_1\le b_1,a_2<X_2\le b_2 \}= F(a_1,a_2)+F(b_1,b_2)-F(b_1,a_2)-F(a_1,b_2)$$ Here $F(a,b)=P\{X_1\le a,\ X_2\le b\}$, the cumulative ...
0
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1answer
20 views

Proving that x $\in \overline{\lim}(A_{n}$ implies for each N there exists at least one n$\geq$N

Letting {A$_{n}$} be a sequence of sets. We are given the definition $$\overline{\lim}(A_{n})= \cap_{n\geq 1}\cup_{k\geq n}A_{k}$$ The question wants us to prove that x $\in\overline{\lim}(A_{n})$ ...
2
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1answer
59 views

Uniformly integrable implies integrable?

The term "uniformly integrable" sounds (to a layman like me) to be stronger than integrable. Just like how uniformly convergent is stronger than simply being convergent. However, from the definition ...
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1answer
35 views

Does there exist a sigma-algebra $F$ such that $f$ is $F/\mathbb{B}$ measurable if and only if $f$ is a constant function?

Let $f$ be a function from $(\mathbb{R},F) \rightarrow (\mathbb{R},\mathbb{B})$ where $\mathbb{B}$ denotes the borel sigma algebra. Does there exist a sigma-algebra $F$ such that $f$ is ...
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0answers
71 views

A confusing question in Royden's Real Analysis

Let $I $ be a closed and bounded interval, and $E $ a measurable subset of $I$. Let $\epsilon>0$. Show that there is a step function $h $ on $I $ and a measurable subset $F $ of $I $ for which ...
1
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1answer
43 views

Definition of essential maximum and essential supremum

Let $\Omega \subset R^n$ a bounded domain. Consider $u \in L^1(\Omega)$. The definition of essential supremum is well known. I am reading the book "Linear and quasilinear elliptic equations" of ...
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1answer
32 views

a countably additive function which is not a measure

Is there a simple example showing that a non-negative countably additive function on a $\sigma$-algebra fails to be a measure?
0
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1answer
31 views

A closed formula for the measure of the union of $n$ sets

Suppose $\mu$ is a finite measure on $(X,\mathscr{A})$ and {${A_n}$}$_{i=1}^n$ is an arbitrary sequence of $\mathscr{A}$-measurable sets. I conjecture that $$ \mu\left(\bigcup_{i=1}^n{A_n}\right) = ...
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2answers
68 views

Do (the restrictions of) continuous maps (to the rationals) form a Borel set?

Consider, within the Polish space $\mathbb{R}^\mathbb{Q}$ (with product topology), the subset of all those maps that can be extended to a continuous map on all of $\mathbb{R}$. It's easy to see that ...
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0answers
40 views

Smallest $\sigma$-algebra containing left-close intervals [duplicate]

I'm trying to solve this task: Let $E:=[0,1[$ , $n\in\{1,2,...\}$ and $0=:a_0<a_1<\cdots<a_n:=1$ Give the $\sigma(C)$, the smallest $\sigma$-algebra on $E$ which contains all elements ...
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1answer
48 views

Exercise measure theory: properties of a function

Let $\Omega \subset R^n$ ($n\geq 2$) a bounded domain. Let $u \in L^{1}(\Omega)$. For each number $c >0$ define $A_c :=\{ x \in \Omega; u(x) > c\}$. Define $$ f(k) := \int_{A_k} (u(x)-k) dx$$. ...
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0answers
34 views

Poisson Point Process - random set and random measure formalism duality

I have a seemingly trivial question but it does concern the formalism duality in point process theory, which I appreciate could be leading to my own confusion. Let $N$ denote a (non-negative ...
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1answer
20 views

If the probability measure $\mu$ is a left-eigenvector to the eigenvalue $1$ of a stochastic matrix $p$, then $\mu p^n=\mu$

Let $E$ be an at most countable set and $\mathcal E$ be the discrete topology on $E$ $p=\left(p(x,y)\right)_{x,y\in E}$ be a stochastic matrix $\mu$ be a probability measure on $(E,\mathcal E)$ ...
1
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1answer
23 views

Integrable of composion of functions

Let $G : \mathbb{R}\rightarrow \mathbb{R}$ is a continuous function such that $|G(t)|/|t|$ is not bounded. Prove that there is a function f belong to $L^p(\mathbb{R})$ (here we use lebesgue measure) ...
1
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1answer
88 views

Show $\mu$ is absly cts w.r.t. $m$

I am working on some practice questions for mt, and am struggling a bit with the last few parts of this question: Let $\Omega$ be a $\sigma$ of subsets of $X$ and let $\mu$, $m$ be finite ...
0
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1answer
34 views

suppose we have a collection $F$ and a countable subcollection $G$ of $F$. suppose $\sigma(G)$ is the sigma algebra generated by $G$. prove:

suppose we have a collection $F$ and a countable subcollection $G$ of $F$. suppose $\sigma(G)$ is the sigma algebra generated by $G$. prove the following statement: the union of all sigma algebra's ...
0
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3answers
134 views

If f is integrable, is it finite almost everywhere?

If $\int_\Omega f d\mu<\infty$, and $f$ is non-negative, can we conclude that $f$ is finite a.e. on $\Omega$? Is being finite a.e. the same as having a finite essential supremum, i.e. there exists ...
0
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1answer
41 views

Modified stochastic processes

I am looking for conditions such that a process $(X_t)_t$ where the $X_t$ are $\text{iid}$ such that there is a process $(Y_t)_t$ satisfying $P(X_t=Y_t)=1$ and $t \mapsto Y_t(\omega) \text{ is ...
3
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1answer
47 views

Find the Lebesgue Measure of the following sets

Find the Lebesgue Measure of the following sets : $A=\{0<x\leq 1:x\sin (\dfrac{1}{x})\geq 0\}$ $B=\{0<x\leq 1:\sin(\dfrac{1}{x})\geq 0\}$ In order to find Lebesgue Measure of a set we ...
2
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1answer
36 views

The necessity of non-negativity for the inequality $\sum_{k\in\Bbb Z^n}f(k+y) \ge \int_{\Bbb R^n} f(x)\,dx$

I'm working through a proof for Minkowski's convex body theorem and there is a short technical lemma relating the sum of an integrable function $f$ over a translate of the integer lattice and its ...
6
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3answers
379 views

Why is this function measurable?

If $f$ is Borel measurable on $\mathbb{R}$, why is then $f(x-y)$ Bore-measurable on $\mathbb{R}^2$? I tried showing that the set $\{(x,y): x-y \in f^{-1}(a,\infty)\}$, is measurable, we know that ...
3
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0answers
74 views

Equivalency of two forms of Lebesgue measurability

Following is the Carathéodory's criterion for Lebesgue measurability $A\subset \mathbb{R}$ is Lebesgue measurable $\textbf{iff}$ for any set B (measurable or not) $m_*(B)=m_*(B\cap A)+m_*(B -A)$. ...
0
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0answers
28 views

limit function and integrals

This time I'm asking you for a revision of an excercise I solved. The problem is: Let $(X,\mathcal{A},\mu)$ a finite measure space. Take $0\le f\in\mathcal{L}^1(X,\mu)$. If $f_n(x)=f(x)^{1/n}$, ...
0
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1answer
76 views

Borel $\sigma$-algebra of continuous functions

Let $B(C(T,\mathbb{R}))$ be the Borel sigma algebra of continuous functions mapping from the compact metric space $T$ to $S$ defined by the canonical metric $\|\cdot\|_\infty.$ Now I was wondering ...
7
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1answer
86 views

$L^q(X) \subset L^p(X)$ if $p\leq q$ and $\mu(X) < \infty$.

Let $(X,\mathfrak{M},\mu)$ be a measure space. Show that if $\mu(X) < \infty$ then $L^q(X) \subset L^p(X)$ for $1\le p \le q \le \infty$. Is it enough to define $$E_0 = \{x \in X \, : \, 0 \leq ...
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0answers
23 views

Heuristic statement of the monotone class lemma

From Folland's Real Analysis: Modern Techniques and Their Applications we have: The Monotone Class Lemma (MCL): If $\mathscr{A}$ is an algebra of subsets of $X$, then the monotone class ...
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2answers
75 views

Characterization of smallest $\sigma$-Algebra on $\Omega=[0,1)$ that contains disjoint intervals of the form $[a,b)$ where $0<a<b<1$

Problem: Let $\Omega$=[0,1) and $0=a_0 < a_1< \dots < a_n =1$ where $n \in \mathbb{N}$ Let $X:= \lbrace [a_{i-1}, a_i) : i =1,2, \dots , n \rbrace$ and find an identity for $\sigma(X)$ ...
2
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0answers
28 views

Showing $\left\lbrace x: \lim f_n(x) \text{ exists} \right\rbrace$ is measurable

Assuming $\left\lbrace f_n\right\rbrace$ is measurable for all n, I was able to show using the fact that sequences of real numbers converge iff they are Cauchy that: $\left\lbrace x: \lim f_n(x) ...
2
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2answers
126 views

Does a sub-$\sigma$-algebra of a countably generated $\sigma$-algebra have to be countably generated?

Assume that $A \subseteq B$ are $\sigma-$algebras and $B$ is a countably generated (separable) $\sigma-$algebra. Now my question: Is it possible that $A$ is not countably generated? I'm ...
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1answer
25 views

Measurable functions of finite measure spaces

Let $(X;\mu )$ be a finite measure space; $f : X \to R$ be a measurable function. Assume $\mu(f > 0) > 0$. Is it true that for every $\epsilon> 0$ there is a $\delta > 0$ such that $\mu(0 ...
8
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1answer
97 views

What's the intuition behind the direct integral of a family of Hilbert spaces?

In order to understand better the mathematically rigorous version of Dirac's formalism in Quantum Mechanics I've been reading about direct integrals of Hilbert spaces, projector-valued measures and so ...
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1answer
41 views

Real analysis about measurable set and Riemann Integrable

We know for $ \forall x \in [0,1], \exists \{a_n(x), n \in \mathbb{N} \} \subset \{0, 1\}$ with $ x = \sum_{n=1}^{\infty}\frac{a_n(x)}{2^n}$. (''binary expansion''). Show that $\{x \in [0,1]: ...
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1answer
27 views

How to write convergence in measure as union and intersections

I know that if $f_{n} \to f$ $\mu$-a.e., then we have that: $$\mu\left( \bigcup_{k} \bigcap_{N} \bigcup_{n\geq N} \{x: |f_{n}-f|\geq 1/k\} \right) = 0$$ Now I want to write something similar for ...
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2answers
152 views

Second derivative of $\int_\mathbb{R}\cos(tx)dp(x)$

Let $p$ be a probability on $\mathbb{R}$ and $$f(t):=\int_\mathbb{R}\cos(tx)dp(x).$$ I want to show that if $f''(0)$ exists then $$f''(0)=\lim_{t\to 0}2\frac{f(t)-1}{t^2} \: \:(\star).$$ By ...
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1answer
36 views

Question about aProof using simple functions.

Im confused about the following proof: why is $E_j$ and $F_k$ the union of intersections? Why is the union of $E_j$ and $F_j$ equal to $X$?
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1answer
24 views

If I have that $Z$ is a random variable, and $f$ is a measurable function, how can I show that $E(f(Z)Y |Z) = f(Z)E(Y|Z)$?

I have that $Z$ is a random variable, and that $f$ a measurable function, and would like to show that: $$ E(f(Z)Y |Z) = f(Z)E(Y|Z) $$ This was under wikipedia's expectation page under the "pulling ...
4
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1answer
111 views

How to show density of 2^a 3^b

Sounds like a nice homework problem, but this actually came up in preparing a lecture for a Music class; I want to show that if you try to build a set of notes where you can go up and down octaves and ...
17
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3answers
632 views

Operator $T \colon L^p \to L^p$ is a conditional expectation

I'm trying to solve this problem: Let $(X,\mathcal{B},\mu)$ a probability space and $T \colon L^p(\mu) \to L^p(\mu)$ a continuous linear operator ($1 \leq p < \infty$ ) with the following ...
1
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1answer
46 views

Continuity and almost everywhere convergence

(Folland 2.37) Suppose that $f_{n}$ and $f$ are measurable complex-valued functions, and $\phi: \mathbb{C} \to \mathbb{C}$. If $\phi$ is continuous and $f_{n} \to f$ a.e., then $\phi \circ ...
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2answers
86 views

Lebesgue integration in $\mathbb{R}^{n}$: Folland 2.56

I am trying to solve this question in Folland's Real Analysis: Modern Techniques and Their Applications, but cannot get anywhere with it: If $f$ is Lebesgue integrable on $(0,a)$ and $g(x) = ...
2
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1answer
60 views

The Scorza-Dragoni theorem as a consequence of Egorov's theorem?

Scorza-Dragoni theorem (at least the version I have used) says that if you have a function $f : \Omega \times \mathbb{R}^{N} \longrightarrow \overline{\mathbb{R}}$ which satisfies: i) $x \rightarrow ...
4
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1answer
79 views

Is total variation a continuous map from complex measures to positive measures?

The following question arises naturally in my current research. It seems to be a basic problem in measure theory, and therefore I guess that answers to it can be found in some textbooks. However, I ...
2
votes
2answers
55 views

Strong Sobolev inequality

Take $B(0,1)$ the ball in $\mathbb{R}^2$ with the normalized Lebesgue measure $\lambda$ such that $\int_{B(0,1)} d \lambda=1.$ Now, I want to show, or give a counterexample that this is false, that ...
3
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0answers
61 views

Measurable function in $L^2$-norm

Let $f$ be a measurable function. Assume you know that $$\sup_{||g||_2=1} \left|\int fg\,\right|$$ exists. Does this mean that $$\sup_{||g||_2=1} \left|\int fg\,\right| = ||f||_2?$$
9
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1answer
244 views

Product of metric outer measures

The problem below has been asked recently already but, as a naive user, I got burned (well singed perhaps) because I asked the question in the wrong place. So if this looks like a redundant question ...
4
votes
1answer
86 views

A function $f$ that is not in any $L^p$ but the measure of $\{|f|>t\}$ is bounded by $C/t$

How can I find a function $f$ such that $f \notin L^{p} (\mathbb{R})$ for all $p$ but you can find a constant $c>0$ for it with $m(x \in \mathbb{R} \, s.t. |f(x)|>t) \leq \frac{c}{t}$ for ...
0
votes
2answers
27 views

Borel measurable function and sets.

Let $f,g : \mathbb{R}^{n}\to [0,+ \infty] $ 2 Borel measurable functions. Show that {${x \in \mathbb{R}^{n}| f(x)<g(x) } $} is a Borel set. I don't know how to start with the problem. At the ...
0
votes
1answer
43 views

Measure that cannot be induced by a pre-measure

I was wondering if there are measures that cannot be constructed (induce by) from a premeasure. Since we are shrinking the algebra to sigma-algebra, some measures like those should be out there. Could ...