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

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1answer
159 views

If $E\subset\mathbb R^d$ has positive measure, then the set $E-E=\{x-y:x,y\in E\}$ contains an open Ball $B_{\epsilon}(0)$

If $E\subset\mathbb R^d$ has positive measure, then the set $E-E=\{x-y:x,y\in E\}$ contains an open Ball $B_{\epsilon}(0)$ with radius $\epsilon$ around the origin $\textbf {Hint}:$ ...
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1answer
66 views

Borel $\sigma$-algebra of the product is the product of the Borel $\sigma$-algebras with $\sigma$-compactness

Suppose $X_1$ and $X_2$ are Hausdorff, locally compact, $\sigma$-compact spaces. Clearly the same holds for their product $X=X_1\times X_2$. We know that in general the Borel $\sigma$-algebra of the ...
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2answers
45 views

Show, that $E(S_T)=E(T)E(X_1)$. [closed]

Let $(X_i)$ be a sequence of independent, identically distributed and integrable random variables on the probability space $(\Omega,\mathcal{A},P)$. Moreover, let $T\colon ...
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1answer
68 views

How's a semialgebra actually a “semi”-algebra?

According to the definitions that I'm familiar with, a semialgebra of a set $X$ defined as a collection $S \subset\mathcal{P}(X)$ , such that: $\emptyset,X\in ...
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1answer
147 views

Prove that there exists a sequence of compact sets $K_1\subset K_2\subset…\subset A$ such that $\mu(A-\cup_{j\ge1}K_j)=0$.

Let $A\subset\mathbb{R}^n$ be measurable. Prove that there exists a sequence of compact sets $$K_1\subset K_2\subset...\subset A$$ such that $\mu(A-\cup_{j\ge1}K_j)=0$. Here $\mu(A)$ is the ...
2
votes
3answers
565 views

Volume of $n$ dimensional ellipsoid

Let $c_1,c_2,...,c_n$ be positive constants. Consider the $n$ dimensional ellipsoid given by $\{(x_1,...,x_n)|\sum_{k=1}^n\frac{x_k^2}{c_k^2}<1\}$. Prove that it's $n$ dimensional volume is ...
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1answer
122 views

Functional Analysis - Finding the multiplicative operator norm over L1

Let $f \in C([0,1])$, the space of continuous real-valued functions over $[0,1]$. Let $\Gamma_f: L^1([0,1],m) \rightarrow L^1([0,1],m)$, the space of complex-valued functions Lebesgue integrable ...
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2answers
85 views

Borel function which is not continuous (in every point)

Give example function $f: \mathbb{R} \rightarrow \mathbb{R}$ which $\forall x \in \mathbb{R}$ is not continuous function but is Borel function. I think that I can take $$f(x) = \begin{cases} 1 & ...
3
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1answer
90 views

Showing almost sure equality of r.v.s in L1 given conditional expectations

Let $X$, $Y$, and $Z$ be three random variables in $L^{1}(\Omega, \mathcal{F}, P)$. Suppose that we have $E(X|Y)=Z$, $E(Y|Z)=X$, and $E(Z|X)=Y$. Show that $X=Y=Z$ a.s. I am able to show this in ...
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1answer
162 views

Prove that if $f(x)$ is measurable function then $h(x)=…$ is also measurable function.

Prove: If $f: ( X, \mathcal{A}) \rightarrow \mathbb{R}$ is measurable function, $A \in \mathcal{A}$ then function $h:( X, \mathcal{A}) \rightarrow \mathbb{R}$ such that $$h(x) = \begin{cases} f(x) ...
3
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0answers
26 views

Weak law of large numbers ($\mathfrak{L^2}$-version)

Let $(X_n)_{n\in\mathbb{N}}$ be a sequence of pairwise uncorrelated random variables out of $\mathfrak{L}_{\mathbb{P}}^2$ on the probability space $(\Omega,\mathcal{A},\mathbb{P})$ with $$ ...
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0answers
25 views

Questions about real-valued measure on a vector space.

I am reading the lecture notes on representation theory. I have some difficulty in solving Exercise 1.8 on page 4. Let $K$ be a non-archimedian local field and $v : K \to Z \cup \{\infty \}$ a ...
3
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2answers
72 views

Deducing that $f$ is in $L^p$

Suppose $f : \mathbb{R} \to \mathbb{R}$ is integrable and there is positive $K$ and $0<c<1$ such that $$\int_{B} \left\vert \:f(x)\right\vert \:\mathrm{d}x \leq Km(B)^{c}$$ for every Borel ...
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2answers
42 views

simple functions

let $m_f(A)=\int_{A} f dm$ suppose that f is a simple function that is $f= \sum c_{i} 1_{A_{i}}$ describe f. this is part of a problem where i had to prove that $m_f(A)$ is a measure and i had also ...
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0answers
37 views

Converse of Kolmogorov's Zero-One Law

The Kolmogorov zero-one law says that for a sequence of independent events, any event belonging to the tail $\sigma$-field has probability either $0$ or $1$. However the converse is not true, because ...
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2answers
151 views

$f^2$ integrable and $f$ is not

I'm trying to find an example of a function that is not Lebesgue integrable but $f^2$ is integrable. The problem I am trying to solve includes the converse for which i gave: $\frac{1}{\sqrt{x}}$ or ...
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0answers
43 views

Proof of Lebesgue Density Theorem [duplicate]

If E is a Borel set in $\Bbb R^n$, the density $D_E (x)$ of $E$ at $x$ is defined as $$D_E(x) = \lim_{r \to 0} \frac{m(E \cap B(r,x))}{m(B(r,x))}$$ Show that $D_E(x) = 1$ for almost everywhere $x ...
3
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1answer
215 views

Convergence in norm!

Let $f_n(x)=\sqrt{n}e^{-nx}$ for $x\in \Omega=(0,1).$ Question: Does $\int_{\Omega}f_n(x)h(x)dx\rightarrow 0$ as $n\rightarrow \infty$ for each $h(x)\in L^1(\Omega)?$. My intuition is that this is ...
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0answers
43 views

Integrability and product measure

Let $X$ and $Y$ be subsets of $\mathbb{R}$, and let $\mu$ be a measure on $X$ and $\nu$ a measure on $Y$. Let $f : X \times Y \rightarrow \mathbb{R}$ be $\mu$-summable and $\nu$-summable, i.e. ...
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1answer
96 views

Variance With Martingales Problem - Answered; Ignore the Bounty

Let $(X_{j})_{j \geq 1}$ be random variables such that $X_{j}$ is $\mathcal{F}$-measurable for each $j$, where $(F_{j})_{j\geq 1}$ is an increasing sequence of $\sigma$-algebras. Assume ...
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0answers
38 views

An $n-1$ dimensional surface has $n$ dimensional measure $0$.

How does one show this? I was thinking that on an $(n-1)$ dimensional surface there a local homeomorphism to $\mathbb{R}^{n-1}$, which can be canonically embedded into $\mathbb{R}^n$, and it seems ...
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0answers
57 views

Improper integral does not exists -> not Lebesgue-measurable?

If I have the measure space $(\mathbb{R},\mathcal{B},\lambda)$ with the Lebesgue-measure $\lambda$ and the improper Riemann-integral of $\lvert f\rvert$ does not exists. Is then $f$ not ...
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1answer
44 views

Claim in Stein and Shakarchi Real Analysis, made stronger

In Stein and Shakarchi, Real Analysis textbook they claim that "The main use of good kernels was that whenever $f$ is bounded then $(f*K_\delta)(x)\rightarrow f(x)$ as $\delta\rightarrow 0$ at every ...
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2answers
47 views

Example of an integral not converging

Consider a measurable space $(\Omega,\mathcal{A},\mu)$ with $\mu(\Omega)<\infty$. Let $f_1,f_2,\ldots$ be bounded measurable functions so, that $f_n\to f$ uniformly. Then $f$ is measurable ...
4
votes
2answers
56 views

How to show that $\{(a,a) : a \in A\}$ is a null set of $A \times A$?

Let $A \subset \mathbb{R}^n$. How do i show that $\{(a,b) : a,b \in A \text{ and } a=b\}$ is a null set of $A\times A$? It is easy to see for intervals but the general case I don't know.
3
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1answer
33 views

Integral over two sets when the measure of their symetric difference equals zero

Let $A$ and $B$ be two measurable sets of $X$. Show that if $m(S(A,B))=0 $ then $$\int_{A} f\, dm =\int_{B} f\,dm,$$ where $S(A, B)$ is the symetric difference. my method was to prove that ...
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1answer
48 views

Transforming a series to an integral with respect to counting measure

I'd really appreciate it if somebody could help me understand why we have this with a step-by-step explanation (i.e. in an argument complete way) : $$ \sum_{k=1}^{n} {\frac {n} {k^2+nk+1}} = \int ...
2
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1answer
118 views

invariant measure under irrational rotation on $S^1$

Prove that if $T:S^1 \to S^1$ is an irrational rotation, then the only probability measure on $S^1$ that is $T-$invariant is the lebesgue measure or a multiple or it. We are considering the ...
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1answer
98 views

If $g$ is invariant under an ergodic map then it's almost everywhere constant

Let $(X,B(X),\mu)$ be a probability space, where $X$ is compact metrizable, and $B(X)$ are the Borel sets. Let $f:X\to X$ be a measurable function such that: i) $\forall A\in B(X)$ ...
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1answer
54 views

Sufficient and necessary condition for Lebesgue integrability of a random variable

Could anyone give me a hand with the following problem? Let $f$ be a random variable over a probability space $(\Omega,A,\mathbb P)$. Show that $f$ is integrable $\iff $ ...
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1answer
46 views

Convergence of $\int_{0}^{2}{k^{p}x^{k} \over 1 + x^{2k}}\,{\rm d}x\quad\mbox{with}\quad k = 1,2,3,\ldots\quad\mbox{and constant}\quad p$

Consider the integrals $$ \int_{0}^{2}{k^{p}x^{k} \over 1 + x^{2k}}\,{\rm d}x\quad\mbox{with}\quad k = 1,2,3,\ldots\quad\mbox{and constant}\quad p. $$ For what values of $p$ do the integrands have an ...
2
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1answer
29 views

measurable sets [duplicate]

please could you help me with this exercise. Prove or a counterexample to the following statement: $f:\mathbb{R} \to \mathbb{R} $ such that $\forall \alpha \in \mathbb{R}$ the set $\{ x \in ...
2
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0answers
76 views

Lebesgue integral for a non-negative, measurable and bounded function

Consider the measurable space $(\Omega,\mathcal{A},\mu)$. Let $f$ be a measurable, non-negative and bounded function. Show that the $\mu$-integral of f is given by $$ \int f\, ...
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0answers
66 views

Question on Sigma Algebras

I have the measure spaces $(\mathscr{X},\mathscr{A},\mu_1), (\mathscr{Y},\mathscr{B},\mu_2), and(\mathscr{Z},\mathscr{C},\mu_3)$. I take the $\sigma$-algebra ...
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1answer
25 views

If $\mathcal{N}=\{(-b,\infty),(a,\infty): a,b\in\mathbb{R}\}$, then ${\mathcal{A}}_{\sigma}(\mathcal{N})=\mathcal{B}(\mathbb{R})$

Let $$\mathcal{N}=\{(-b,\infty),(a,\infty): a,b\in\mathbb{R}\}$$ I want to show that the $\sigma$-algebra generated by a family of $\mathcal{N}$, is the family of Borel sets, ...
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1answer
22 views

Existence of a Set s.t. the Integral of a Non-negative Function Concentrates on It

Let $(X,\mathcal{M},\mu)$ be a measure space. Show that if $f\in L^+$ (measurable and non-negative functions) and $\int fd\mu<\infty$ then, $\forall \varepsilon >0$ $\exists E\in\mathcal{M}$ ...
2
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1answer
163 views

counterexample for Dominated Convergence Theorem

The Dominated Convergence Theorem is as follows: What if the sequence $\left\{f_n \right\} \notin L^1$? Could someone provide a counterexample as to why the theorem wouldn't hold? Thanks!
2
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1answer
76 views

There exists no continuous function $f:\mathbb{R}\rightarrow\mathbb{R}$ with $f=\chi_{[0,1]}$ almost everywhere [duplicate]

I am trying to solve the same problem on this page. One gave an hint defining an inclusion function. Does someone know what is meant there? Thanks
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1answer
51 views

Mean square integrable

Let $f:$ $X \rightarrow R$ be a measurable function such that f is mean square integrable. (i.e $ \int f^2 dm< \infty$). Show that if $ m(X)< \infty$ then f is integrable I tried to show that ...
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2answers
59 views

E(X|X+Y) X,Y iid exp(-1)??

How do you find the random variable E(X|X+Y) where X, Y are i.i.d exponential random variables with λ = 1?
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1answer
33 views

Continuity/Differentiability of Fourier Series

Possibly stupid question: I'm wondering if there is some trick for evaluating the continuity/differentiability of a Fourier series. In particular, I'm looking at the function $f(x)=\sum_{n=0}^\infty ...
2
votes
1answer
91 views

Do The Integrals Tend to 0?

Consider the integrals $\int_1^\infty \frac{k}{x^2+k^p\cos^2x}dm(x),$ where $m$ is the Lebesgue measure. For what $p$ do the integrands have an integrable majorant? For what $p$ do the integrals tend ...
2
votes
1answer
42 views

Distribution of Difference of Independent Random Variables

Usually in the development of the theory of Brownian motion, one makes the assumption that $X_t$ (the coordinate functions on $(\mathbb{R}^*)^{[0,\infty)}$). have normal distributions with mean $0$ ...
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2answers
142 views

Simple explanation for a Borel Set

I am from engineering background and the concept of Borel set, Borel field and measures sound abstract to me. Can some one please, explain them in a simplified way i.e., to explain them without ...
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0answers
87 views

Regular Borel measures on $\mathbb{R}$ and right continuous increasing functions $\mathbb{R}\to\mathbb{R}$

I'm reflecting Borel measures on $\mathbb{R}$. I know the following two things Let $F:\mathbb{R}\to\mathbb{R}$ be a right continuous, increasing function. Then we look at the algebra $H$ of finite ...
1
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2answers
137 views

True or false: Exchanging this limit and sum

True or false: For any $a > 0$: $\displaystyle \lim_{x \to 1} \sum_{n=0}^{\infty} x^n e^{ - a n} = \sum_{n=0}^{\infty} e^{- a n} = \frac{1}{1 - e^{-a}}$ I know when we can exchange limits and ...
3
votes
1answer
142 views

Evalute $\int_{0}^{\infty} \frac{\sin 4x}{1 + x^4} dx$

The problem is to evalute $$ \int_{0}^{\infty} \frac{\sin 4x}{1 + x^4} dx. $$ It may be too long to write all the details of the solution here. It will probably suffice for me to know what kind of ...
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0answers
57 views

Estimate Borel Sets with Open Sets

We're given the measure space $(X,\mathscr{A}, \mu)$with $X=\bigcup_{i=1}^\infty X_i$with $X_i\subset X_{i+1}\subset ...$, $X_i$are open for all i and $\mu(X_i)<+\infty\space\space\space\space ...
1
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0answers
48 views

Counterexample t0 prove that B is not a sigma algebra

Let $f: \Omega \rightarrow E$ and let $A$ be a $\sigma$-algebra of $\Omega$. Let $B = \{y \in E: \exists x \in A$ such that $f(x)=y \}$. We have to prove that $B$ is not a $\sigma$-algebra. Is the ...
3
votes
1answer
57 views

Is there a Peter-Weyl theorem for the quasi-invariant measure on a homogeneous space of a compact semisimple group?

Let $H \hookrightarrow G$ be an inclusion of semisimple, compact Lie groups. There is a measure on the homogeneous coset space $G/H$ by pulling back the Haar measure on $G$ via the projection $G ...