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

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1answer
17 views

Prove that $\sigma (\cap_{i \in I} C_i)=\cap_{i \in I} \sigma (C_i)$

Do we have the following identity? $$\sigma (\cap_{i \in I} C_i)=\cap_{i \in I} \sigma (C_i)$$ Here $C_i$ is a subset of a set $\Omega$.
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1answer
31 views

Integration with respect to a signed measure

Let $\mu$ be a singed measure, $f\in C_c(X)$, I want to show $$\int fd(c\mu) = c \int fd\mu, \forall c \in \mathbb{R}$$ Since $c\mu$ is also a singed measure, I think by definitionm I need to show ...
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1answer
24 views

Show that an irregular 1-set in the plane is totally disconnected

A $1$-set is a Borel set such that $0 < \mathcal{H}^1(A) < \infty$, where $\mathcal{H}^s$ is the Hausdorff measure. Let $A$ be an irregular $1$-set in the plane. Deduce from the theorem below ...
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1answer
25 views

$\varphi(x)=\int_{[\xi_0,\xi]}f(x+t)d\mu_t$ absolutely continuous and summable on $\mathbb{R}$

Let $f\in L_1(-\infty,\infty)$ be a Lebesgue-summable function on $\mathbb{R}$. I read that the function$$\varphi(x)=\int_{[\xi_0,\xi]}f(x+t)d\mu_t$$is absolutely continuous on any real closed ...
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0answers
24 views

Special Lebesgue measurable subsets of $\mathbb{R}$ [closed]

Construct a subset $A$ of $\mathbb{R}$ which satisfies following properties: (a) $A$ is closed (b) $A$ is Lebesgue measurable (c) $m(A) > 0$ ($m(A)$ is the Lebesgue measure of $A$) (d) $A$ has ...
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1answer
43 views

Computing a Projection Valued Measure

I've recently begun learning about Projection Valued Measure and I'm a little confused. I understand that a Projection Valued Measure is a family of orthogonal projections $P(\Lambda)$ indexed by the ...
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1answer
25 views

Cartesian Product of Borel Sets is Borel Again

Let $E$ and $F$ be Borel measurable subsets of $\mathbb R^{d_1}$ and $\mathbb R^{d_2}$, respectively. Then $E \times F$ is also Borel measurable in $\mathbb R^{d_1 + d_2}$. I suppose it is ...
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1answer
28 views

Construction a measure in $[0,1]$

Suppose $\epsilon \in (0, 1)$ and $m$ is Lebesgue measure. Find a measurable set $E \subset [0, 1]$ such that the closure of $E$ is $[0, 1]$ and $m(E) = \epsilon$. Having trouble on where to start
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2answers
15 views

Proving that a specific linear functional exists.

If $m$ is Lebesgue measure on $I=[0,1]$ with $L^\infty=L^\infty(m)$, how might I go about showing that there is a bounded linear functional $\Lambda\neq0$ on $L^\infty$ that is $0$ on $C(I)$? (This ...
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0answers
26 views

Construction of measure on open and closed sets for $\epsilon>0$

This is from a book which supposed to help you prepare for entrance exam. I am currently on a chapter regarding construction of measure and Caratheodory theory. I am having quite a hard time ...
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2answers
40 views

Cauchy Schwarz Inequality with probability density function!

$$\left\lvert\int_a^b x(t) y^{\ast}(t) p(t) dt\right\rvert^2 \leq \int_a^b |x(t)|^2 p(t) dt \int_a^b |y(t)|^2 p(t) dt$$ where $p$ is a PDF: $p(t)\geq 0$, almost everywhere in $[a,b]$ and $\int_a^b ...
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1answer
27 views

If $g$ nonnegative has bounded support and $\int g^2 d \lambda$ is finite then $\int g d \lambda$ is finite

If $g$ nonnegative has bounded support and $\int g^2 d \lambda$ is finite then $\int g d \lambda$ is finite Previous question asked you to prove Markovs inequality so I think it may have ...
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0answers
48 views

Counterexample to converse of Fubinis theorem

Given $f(x,y)=\dfrac{xy}{(x^2+y^2)^2}$ and the Gaussian measure $\mu$ with expected value 0 and variance 1, show that ...
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1answer
15 views

Composition of monotonic and measurable function is measurable

I am trying to find a counterexample since I think monotony is not enough. Let $g:\mathbb{R} \to \mathbb{R}$ be monotone and for all $n \in \mathbb{N}, f: \mathbb{R}^n\to \mathbb{R}$ is measurable. ...
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1answer
22 views

Pointwise convergence implies mean square convergence on a finite length space.

I am trying to prove that pointwise convergence implies mean square convergence on $[-\pi,\pi]$. I ma not even sure this is true but (my measure theory classes) have shown me that it is. However I am ...
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1answer
45 views

$\mu$ measure on $\cal B_{\mathbb R}$ that is“linear” for some $c\in\mathbb R$ , $\mu (E)=c\cdot m(E)$

i need to show that for all bounded $E\in\cal B_{\mathbb R}$ (while $\cal B_{\mathbb R}$ is the borel $\sigma$-algebra) and for $\mu$ a measure on $\cal B_{\mathbb R}$ and $\mu (E)<\infty$ and ...
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2answers
44 views

Proofs of properties of a measureable and Lebesgue integrable function

Could I get some help showing the following properties to be true: a) $f: X \to [0,\infty) $ is measurable and $\int f d\mu < \infty$ $\forall a > 0$, let $X_a = \{x \in X :f(x) >a\}$, show ...
1
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1answer
51 views

Computing moments

given $\int_{-\infty}^{+\infty} \! e^{-tx^2} \, \mathrm{d}\lambda x = \sqrt{\pi/ t} $ I have been asked to compute the moments $\int_{-\infty}^{+\infty} \! x^{2n} e^{-x^2} \, \mathrm{d}\lambda x $ ...
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0answers
32 views

Probability measure limit

I came across this old qualifying exam problem and I am stuck. Suppose $\mu$ is a probability measure and $f:X\rightarrow [0,\infty]$ is measurable. Suppose that $$G=\displaystyle{e^{\int \log(f) d ...
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1answer
18 views

Show that the product $\sigma $-algebra is generated by the $\pi $-system $\{B _1 \times B _2 : B \in \Sigma _1 , B _2 \in \Sigma _2 \} $

Show that the product $\sigma $-algebra is generated by the $\pi $-system $\{B _1 \times B _2 : B_1 \in \Sigma _1 , B _2 \in \Sigma _2 \} $ Let $(X , \Sigma _1) $ and $(Y , \Sigma _2 ) $ be two ...
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2answers
31 views

Prove that $(\bigcup _{i=1 } ^{\infty } E _i )_x = \bigcup _{i=1 } ^{\infty } (E _i) _x $

Prove that $\left(\bigcup_{i=1}^\infty E _i\right)_x = \bigcup _{i=1}^\infty (E_i)_x$. where $E _x= \{y : (x,y) \in E \}$ I'm not really content with the following, which is my attempt, ...
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1answer
35 views

Continuous and preserves measurability $\implies$ preserves null sets.

Let $X$ be a (Lebesgue-)measurable set of $\mathbb{R}^n$ and $f:X \to \mathbb{R}^n$ continuous function that preserves measurability ($A$ meausurable $\implies f(A)$ measurable). Prove: for all $A ...
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1answer
31 views

Why does Continuous Partial Differentiability Imply Total Differentiability?

Let $f: \mathbb{R}^d \to \mathbb{R}$ be such that the partial derivatives $\frac{\partial f}{\partial x_i}:\mathbb{R}^d \to \mathbb{R}$ exist everywhere and are continuous. Then show that $f$ is ...
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1answer
31 views

If $X:=(X_1,X_2)$ has density $e^{-(x_1+x_2)}1_{\mathbb{R}_{\ge 0}^2}(x_1,x_2)$, then $X_1+X_2$ and $X_1/X_2$ are independent

Let $X_1$ and $X_2$ be real-valued random variables, such that $X:=(X_1,X_2)$ has the density $$f_X(x_1,x_2)=\begin{cases}e^{-(x_1+x_2)}&\text{, if }x_1,x_2\ge 0\\0&\text{, ...
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0answers
23 views

increasing functions are borelian functions

Let $X : \Omega \rightarrow \mathbb{R}$ a discrete random variable and $f : [0, \infty) \rightarrow (0, \infty)$ an increasing function. Show that $f$ is a borelian function. It suffices to show that ...
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1answer
65 views

Show that the Fourier Transform is differentiable

This might be a silly question. For $f$ an integrable, complex-valued function, its Fourier transform is $$ \hat{f}(s) = \frac{1}{\sqrt{2\pi}}\int_{-\infty}^{+\infty} e^{-isx}f(x)\, \mathrm{d}x $$ I ...
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0answers
22 views

Convex sets in $\mathbb R^n$: Do they have a particular form ? Does the gradient of a linear convex function $f$ exist on such a set?

Convex sets in $\mathbb R^n$: Do they have a particular form ? Does the gradient of a linear convex function $f$ exist on such a set ? Suppose that $S$ is a convex set in $\mathbb R$. Then $S$ is ...
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1answer
21 views

Union of Two Rectangles is the Disjoint Union of at most $6$ Rectangles

Let $X = X_1 \times X_2$ and suppose that $(X_1, \mathcal M_1, \mu_1)$ and $(X_2, \mathcal M_2, \mu_2)$ are two measure spaces. Consider the set of all rectangles, i.e., sets of the form $A \times B$, ...
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1answer
23 views

An example of an infinite absolutely continuous signed measure

I'm trying to find an example to show that the following theorem doesn't hold if $\nu$ is infinite: Let $\nu$ be a finite signed measure and $\mu$ a positive measure on $(X, \mathcal{M})$. Then ...
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1answer
26 views

Almost sure convergence clarification

Let $\Omega=(0,1]$, $\mathbb{P}$ be Lebesgue measure, $n\in\mathbb{N}$ be $n = i + 2^j$, where $j = \lfloor\log_2(n)\rfloor$ and $0 \leq i \lt 2^j$. Let's say I have a random variable ...
1
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1answer
21 views

Two notions of absolute continuity

If ν is a signed measure and µ a positive measure, we say that ν is absolutely continuous w.r.t. µ if µ(E) = 0 ⇒ ν(E) = 0. If |ν| is a finite measure then this ...
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1answer
30 views

Show that there are $(\alpha_m)_{m\in\mathbb{N}}$ and $(A_m)_{m\in\mathbb{N}}$ such that $f(x) = \sum_{m=1}^\infty\alpha_m\chi_{A_m}(x)$

Assignment: Let $f: \mathbb{R}^n \rightarrow [0,\infty]$ be measurable. Show that, there is non-negative sequence $(\alpha_m)_{m\in\mathbb{N}} \subset [0,\infty)$ and a sequence ...
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0answers
66 views

What does the author mean here?

I am reading a paper and there is a short statement by Author for which I am not sure if I got him right or not. Could someone let me know if my understanding right or not? He says the following: ...
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0answers
50 views

Prove g is not integrable on any interval

Q/ Let $f(x)=x^{-\frac{1}{2}}$ for $x\in(0,1)$ and 0 otherwise. Let $r_k$, k=1,2,3...be an enumeration of all rationals and set $g(x)=\sum_{k=1}^{\infty}2^{-k}f(x-r_k)$ Prove $g^2$ is finite almost ...
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1answer
35 views

How to show that this process is a martingale

Consider the probability space $([0, 1), \mathcal{B}[0, 1), \lambda)$, where $B[0, 1)$ are the Borel sets on $[0, 1)$ and $\lambda$ is the Lebesgue measure. Let \begin{align*} I_k^n = \left[ ...
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2answers
29 views

continuous functions are measurable for any $\sigma$-algebra?

A function $f:\Sigma\ni X\to Y$ with a $\sigma$-algebra $\Sigma$ is called measurable if $f^{-1}(A)$ is measurable for all open sets $A\subset Y$. Now let $f:\mathbb R\to\mathbb R\cup\{\pm\infty\}$ ...
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0answers
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Can we have $|\int _{\{f _n < f \} } (f _n - f )d \mu|<|\int _X (f _n - f )^- d \mu | $ where $f _n \to f $ (a.e.)

Can we have $|\int _{\{f _n < f \} } (f _n - f )d \mu|<|\int _X (f _n - f )^- d \mu | $ where $f _n \to f $ (a.e.) For me these two integrals are identical, but I have proof where the ...
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1answer
28 views

Problem with topological proof about borel measures

It is given a finite Borel measure $\mu$ on a polish space $E$. The claim is that $\mu$ is then a regular measure. In the proof, it is shown that for any closed set $A$ it holds$$(1) \quad \mu(A) = ...
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2answers
41 views

The $\sigma$-algebra defined on the space of measures $\mathcal{M}(S)$

In Kallenberg's textbook,he defines a space $\mathcal{M}(S)$ that contains all measures in $S$. In the picture above,he defines a $\sigma$-algebra on $S$ by ...
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1answer
33 views

Sets of measure zero and the Lebesgue differentiation theorem.

This is an exercise from Stein-Sharachi Chap. 3 Exx. 25 $\textbf{Problem Statement}$ Let $E$ be a set of measure zero in $\mathbb{R}^d$. Show that there exists a non-negative integrable $f$ in ...
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1answer
23 views

Improvement of weak type inequality for Hardy-Littlewood Maximal inequality

Let $B(x,R)$ denotes the ball in centered at $x\in \mathbb{R}^n$ with radius $R$. The centered Hardy-Littlewood maximal operator $M$ is defined by \begin{equation} Mf(x)=\sup_{B(x,R)} ...
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0answers
19 views

Covering argument for Hausdorff measures

So there is this proof of a lemma in "Measure theory and fine properties of functions" by Evans which is on page 208. Im am confused at the part where he says "by standard covering argument we get ...
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1answer
29 views

Showing a set of limit points of a sequence of measurable functions is measurable.

I have been wrestling with this question and I am not sure how to solve it. Question: Let $(X, s)$ be a measure space and $\{f_n\}$ a sequence of measurable functions such that $f_n:X\to R$ with the ...
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1answer
23 views

Variation Signed Measure Inequality

Given $\nu_1,\nu_2$ finite signed measures, is there a way to prove $|\nu_1 + \nu_2| \le |\nu_1| + |\nu_2|$ without resorting to the fact that for a general signed measure $\nu$, $$|\nu|(E) = \sup ...
5
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1answer
35 views

Total variation distance of two normal random variables $X_t \sim \mathcal{N}(0,s)$ and $X_s \sim \mathcal{N}(0,t)$

I need to prove that the total variation distance between two normal random variables $X_t \sim \mathcal{N}(0,s)$ and $X_s \sim \mathcal{N}(0,t)$ converges to $0$ when $s \nearrow t$. We know that ...
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1answer
26 views

Find area of unit square using outer Hausdorff-measure

We put $$\eta_{\delta}(E) = \inf\left\{ \sum_{i \in \mathbb{N}} \text{diam }U_i : E \subset \bigcup_{i\in \mathbb{N}} U_i \text{, and diam }U_i\in(0,\delta] \right\}$$ and for $E\subset \mathbb{R}^2$ ...
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0answers
24 views

Suppose $f_n \rightarrow f$ in $\mu$-$p$-mean and $f_n \rightarrow g$ $\mu$.a.e. What can be said about $f$ and $g$?

Let $(X, \mathcal E, \mu)$ be a measure-space, $p \in (0,\infty)$ and $f,g, f_1, f_2, \ldots \in \mathcal M(\mathcal E)$. I've been wondering about the following: Suppose $f_n \rightarrow f$ in ...
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2answers
32 views

Existence of distinct points with rational difference in lebesgue measurable set

Let $X \subset [0,1]$ be Lebesgue measurable with $\mu(X)>0$. Show that there exist two (distinct) points $a, b \in X$ with $a-b \in \mathbb{Q}$. I've thought about this for a while but can't seem ...
0
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0answers
18 views

Basic question about $L_1$ integrable functions [duplicate]

I need to show that if a function is in $L_1$, i.e., the integral of $|f|$ over the real line (Lesbesgue measure) is finite, then the limit as $n$ goes to infinity of the integral of $f$ (from $n$ to ...
1
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0answers
32 views

Product of measure spaces(Completion Problem)

Sorry, If this question is asked before, but i had trouble to find it, the question arises when I see Fubini-Tonelli's theorem for complete measure spaces.. As an assumption, we assume ...