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

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Is Lebsegue Measure Translation Invariant?

I am trying to prove that the Lebsegue measure is translation-invariant. Namely, given a set $X\subseteq\mathbb{R}$, I'd like to show $X + y$ is measurable and $\mathit{m}(X + y) = \mathit{m}(X)$. ...
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0answers
15 views

Conditional expectation with respect to two sigma algebras

So the problem is to define two sigma algebras, a stochastic variable, specify a probabilty measure on the sample space $\Omega$ and show that the following relation doesn't equal: ...
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0answers
22 views

Almost sure bounded imply finite expectation?

Suppose that the random variable $X$ is $\mid X \mid<M$ almost surely, for some constant $M<\infty.$ Then can we say that $E(X)<C$ for some constant $C<\infty$? If the expectation is not ...
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0answers
11 views

An example of convergence to Young measures

$\newcommand{\R}{\mathbb{R}}$ $\newcommand{\lam}{\lambda}$ I am trying to prove the following claim: Let $\{u:[0,1]\to \mathbb{R} \mid u \, \, \text{ is differentiable a.e}, u(0)=u(1)=0 \}^{*} $. ...
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0answers
13 views

Necessary and sufficient conditions (1) rv to density function (2) distribution to rv

(1) Let $(\Omega,\mathcal{F},P)$ be a probability measure space and $X:\Omega \rightarrow \mathbb{R}$ a random variable. Let $P_X,~F_X$ denote the probability measure, pdf induced by $X$, ...
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2answers
27 views

Integral of a measurable function

I do not know what should i keep as title for this question... Question goes like this.. Let $f:\mathbb{R}\rightarrow [0,\infty)$ be a measurable function. If $\int_{-\infty}^{\infty}f(x)dx=1$ prove ...
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2answers
31 views

Is there a probability measure on the Cantor set?

I know that the Lebesgue measure of the Cantor set is $0$. Is there a finite positive regular measure on the Cantor set?
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1answer
24 views

Lebesgue Measure of $A=\left \{ (x,0) : x \in [0,1]\right \} \subset \mathbb{R}^2$

Let $A=\left \{ (x,0) : x \in [0,1]\right \} \subset \mathbb{R}^2$ and $m_2$ Lebesgue Measure of $\mathbb{R}^2$. I want to determine $m_2(A)$. So. I know that Lebesgue Measure of interval is b-a. And ...
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0answers
29 views

Conditions for a function to lie in $L^p(\mathbb{R})$ [on hold]

Let $(X, \mathfrak{M})$ be a measurable space. What are some sufficient and necessary conditions for a function $f : X \to \mathbb{R}$ to lie in $L^p(\mathbb{R})$ for $p \in [1,\infty]$? Is true ...
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0answers
61 views

Are all sets with finite measure measurable?

In my textbook, it says: "Let E be any set with m*(E) < $\infty$. Then E is measurable if and only if there exists a measurable set B with m(B) = m*(E)." There always exists a measurable set of ...
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11 views

Sufficient and necessary conditions for representation of a ordered structure with a binary operation.

Given a structure $\mathcal{A} = (A, \succsim, \sqcup)$, where $A$ is a non-empty set, $\succsim$ is a weak order, $\sqcup$ is a binary operation on $A$, let $\mu$ be an order-preserving mapping from ...
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2answers
40 views

Several questions about Riesz–Markov–Kakutani representation theorem

This is a list of questions about Riesz–Markov–Kakutani representation theorem . 1)If $f\in L^1(\mu)$, is it true that $\phi(f)=\int_Xfd\mu$, where $\mu$ is given by the theorem? I am quite sure it ...
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0answers
21 views

Show $A=\{x\in \Bbb{R}^n|\sum_{j=1}^{n}|x_j|^p\le 1\}$ is Jordan measurable for $p>0$

Show $A=\{x\in \Bbb{R}^n|\sum_{j=1}^{n}|x_j|^p\le 1\}$ is Jordan measurable if $p>0$. I did show it is a bounded set because if there exists $x^{(N)}\subset A $ such that $||x^{(N)}||\to \infty $ ...
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1answer
29 views

Proving the monotonicity of a countably additive set function on a $\sigma$-algebra [on hold]

Let m be a set function defined for all sets in a $ \sigma$-Algebra $\scr A$ ; Assume that m is countably additive over countable disjoint collections of sets in A , with values in $[0,\infty ]$ ...
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3answers
54 views

For $\int f < \infty$, the measure of the set of points where $f=\infty$ is zero.

I fear this question was already discussed here, but I was not able to find it. Please remove if it is a duplicate. Prove: For a function $f\geq 0$, if $\int f < \infty$, then the measure of ...
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3answers
66 views

Show that f is measurable.

Let $a > 0, b \geq 0$ and the function $f: \mathbb{R} \to \mathbb{R}$ $$f(x) = \left\{\begin{matrix} 1, & |x| \leq a \\ b & |x| > a \end{matrix}\right.$$ show that it is measurable. ...
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1answer
38 views

Density of $L^\infty(\Omega)h$ in $L^p(\Omega)$ where $h \in L^p(\Omega)$ [on hold]

Let $(\Omega,\mu)$ be a finite measure space. Suppose $1\leq p <\infty$. Let $h$ be an element of $L^p(\Omega)$ with $h >0$ a.e.. How show that the subspace $L^\infty(\Omega)h=\{ f h\ :\ f\in ...
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1answer
25 views

Measurablity of functions defined over sections of product measures

I have to solve the following exercise but I am unable to proceed. Could you please give me some hints to how to solve it? Let $(\Omega_1, \mathcal{F}_1)$ and $(\Omega_2, \mathcal{F}_2)$ be ...
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1answer
33 views

E Lebesgue Measurable implies E^2 Lebesgue Measurable?

Suppose $E \subset \mathbb{R}$ is Lebesgue measurable. Define $$ E^2 = \{x^2 : x \in E\}. $$ Is $E^2$ Lebesgue measurable as well? I believe the answer is yes, but I am struggling to prove it. I ...
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1answer
29 views

Real Analysis, Folland 3.4.26, Differentiation on Euclidean Space

Background Information - A Borel measure $\nu$ on $\mathbb{R}^n$ will be called regular if i.) $\nu(K) < \infty$ for every compact $K$ ii.) $\nu (E) = \inf\{\nu(U): E\subset U, U \ ...
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1answer
16 views

Distribution function derivative bounds give bounds on associated measures? Billingsley theorem 31.4 proof.

I am working through Billingsley, Probability & Measure. Struggling with the proof of theorem 31.4: Suppose $u(a,b) = F(b) - F(a)$ and that $F'$ exists throughout a Borel set $A$. If $F' ≤ c$ ...
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0answers
52 views

Measure $m=\mu$ if $\int fdm=\int fd\mu$

Suppose $X$ is a locally compact Hausdorff space, $m,\mu$ are two Borel measures, if for any $f\in C_c(X)$, $\int fdm=\int fd\mu$, is it true that $m=\mu$?
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22 views

Defining a measure by positive functional

In big Rudin's book, it constructs the Lebesgue measure by first defining a positive functional, and then using Riesz representation theorem. It arises me to think that if every measure can be ...
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21 views

If K=P(X) , then λ is a pre outer measure if and only if it is an outer measure. [on hold]

If K=P(X) where K is an algebra, then λ is a pre outer measure if and only if it is an outer measure. Is it enough to prove that all sets in K are measurable? Any suggestions.
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18 views

n-1 dimensionnal Hausdorff measure and codimension 1 measure

I've been told that on a n-dimensionnal Riemannian manifold, the Hausdorff measure of dimension n-1 and the codimension 1 measure $v_{-1}$ (defined below) are mutually absolutely continuous. I've ...
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56 views

Limit theorems in measure theory

From probability theory/measure theory we know set of theorems such as Monotone convergence, dominated convergence or conditions like uniform integrability which deals with the general question of ...
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1answer
34 views

Prove that there are at most a countable amount of $x \in X$ with $\{ x \} \in \mathcal{A}$ so that $\mu(x) > 0$.

Let $(X,\mathcal{A}, \mu)$ be a finite measure space so that $\mu(X) < \infty$ prove that there are at most a countable amount of $x \in X$ with $\{ x \} \in \mathcal{A}$ so that $\mu(\{x\}) ...
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23 views

How to compute the Lebesgue-Stieltjes measure for given intervals

Let u be a Lebesgue-Stieltjes measure on the Borel σ-algebra. Let Fu be the associated function such that u([a,b)) = Fu(b)−Fu(a). Calculate a) u([a,b]); in terms of the function Fu. b) u((a,b)); in ...
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1answer
35 views

Prove the following integral is asymptotically zero

I have to solve the following exercise. I would appreciate to get a hint for it. Suppose $(\Omega, \mathcal{F}, \mu)$ be a measure space. Let $f$ be an integrable function. Show ...
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3answers
42 views

Sigma Algebra - Partition

Let $\Omega = \{1, 2, . . . , 7\}$ and let $A = \{\{1, 2, 3, 7\}, \{2, 3, 4, 5, 6\}\}$. Find $P(A)$. P is for Partition. I got $P(A) = \{\{1,4,7\}, \{2,3\}, \{5,6\}\}$ If this is wrong, can you ...
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1answer
33 views

$λ = f dµ$ and $ρ = g dµ$ for Lebesgue measure $µ$, give necessary and sufficient conditions on $f,g$ for $λ ⊥ ρ$ and $λ << ρ$

Le $f,g : \mathbb{R} → \mathbb{R}$ be extended integrable functions. Let $λ = f dµ$ and $ρ = g dµ$ for Lebesgue measure $µ$. Give necessary and sufficient conditions on $f,g$ for $λ ⊥ ρ$ and necessary ...
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1answer
36 views

Subadditivity of the $n$th root of the volume of $r$-neighborhoods of a set

Let $A$ be a closed subset of $\mathbb{R}^n$. For $r>0$, let $A_r$ be the $r$-neighborhood of $A$, namely the set $\{x:\operatorname{dist}(x,A)\le r\}$. Let $f(r) = \mu(A_r)^{1/n}$ where $\mu$ is ...
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7 views

The $\mu^{*}$ measurable set of Riesz–Markov–Kakutani representation theorem

In the proof of Riesz–Markov–Kakutani representation theorem, we define $\mu^{*}(V)=\mbox{inf}\{\mu(U),V\subset U\}$ where $U$ is open, it is quite obvious that such definition gives an outer measure, ...
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1answer
32 views

Showing that $\mu$ is a measure when continuous from above

Statment Let $\mu$ be a set function defined on a $\sigma$ -algebra. Show that $\mu$ is a measure given that $\mu \geq 0$, $\mu(\emptyset)=0$, $\mu$ is continuous from above and countably additive. ...
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27 views

For a stopping time $T$, prove that $X^T_t = \mathbb{E}\left[X_T\mid \mathcal{F}_t\right]$

We have a sigma-algebra $\mathcal{F}=\mathcal{F}_{\infty}$, a stopping time $T$ and an integrable random variable $X$ and define a martingale by $X_t = \mathbb{E}[X \mid \mathcal{F}_t], ...
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1answer
33 views

$\sigma$-algebra generated by weak topology in Hilbert Space

In general, if we have $H$ Hilbert space, and equipped with the weak topology, say $\tau^\ast$, is $\sigma(\tau^*)=\mathcal{B}$?, where $\mathcal{B}$ is the usual Borel $\sigma$-algebra I suspect it ...
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1answer
13 views

Question about Folland's proof of extension-of-premeasures theorem

Here is an excerpt from Folland's Real Analysis. I don't understand why the calculation $\nu (E)\leq \sum _n \nu (A_n)=\sum _n \mu_0(A_n)$ implies $\nu(E)\leq \mu (E)$. Why is this? The $A_n$ are not ...
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1answer
31 views

Local Riesz Potential estimate in terms of Maximal Function

For $f \in L^1_{\text{loc}}(\mathbb R^n)$, and fixed $R > 0$ we defined the local Riesz potential by $$I(x) = \int_{B(x,R)} \frac{f(y)}{\lvert x-y \rvert^{n-1}} d\lambda (y), \hspace{1cm} x \in ...
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Simple random walk on $\mathbb Z^d$ and its generator

I'm still trying to figure out definitions and properties of random walks on $\mathbb Z^d$. My goal is to work up to understanding some large deviation principles for the local times of such random ...
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Categorically deducding measurability of sections

Two lemmas which are often proved in elementary measure theory courses are that sections of measurable sets are measurable, and sections of measurable functions are measurable. Note $E_x= \left\{y\in ...
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1answer
75 views

How can I show that one of $m(A)$ or $m(\Bbb{R}\setminus A)$ is zero?

Let $A \subseteq \Bbb{R}$ be Borel measurable, and $T$ a dense subset of $\Bbb{R}$. Suppose for every $t \in T$ that $$m((A+t)\setminus A)=0,$$ where $m$ is the Lebesgue measure. Then I want to show ...
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1answer
29 views

Borel Sigma Algebra generated by (a, b] [on hold]

Let {(a,b]} be a class of sets, where a and b is an element of R, a < b, a can be negative infinity and b can be positive infinity. Let B be the sigma algebra generated by the class. Show that the ...
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1answer
17 views

If two functions differ on a set of positive measure, must their essential infima differ, too?

Suppose $f,g : [0,1]^2 \to [0,1]$ are measurable functions differing on a set $P$ of positive Lebesgue measure. Claim: there exists $A, B \subseteq [0,1]$, each of positive measure, such that ...
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1answer
37 views

Prove that $\sigma(F)=\Omega$

Let $F=\{A_1,...,A_n\}\subset P(X)$; $F_a=A_1^{a_1}\cap A_2^{a_2}\cap\cdots \cap A_n^{a_n}$ $ a=(a_1,...,a_n)\in \{0,1\}^n$ $$A^{a_i} = \begin{cases} A, & \text{if } a_i=0 \\ A^c, & ...
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0answers
17 views

Borel isomorphism between polish spaces

In my lecture on stochastics the following result has been used: For any uncountable Polish space $X$ there is a Borel isomorphism between this space and the real line. I was not able to find a ...
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1answer
19 views

Proving a variation of DCT

As homework, I was given the following problem. Suppose $f_n\overset{\text{a.e}}{\rightarrow}f$, and for each $n$ there's a $g_n\in L^1$ satisfying $|f_k|\leq g_k$. Prove that if $g=\lim _n g_n$ is ...
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0answers
25 views

any sum of sets open\nullset is a set of the same form

I'm curious how can one prove that any sum of sets $G\setminus N$, where $G$ is open and the Lebesgue measure of $N$ is 0, is a set of the same form. it is easy for countable sums, but in general? ...
2
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1answer
27 views

Does there exists a $\sigma$-algebra $\mathcal{F}$ such that$f$ is $\mathcal{F}/\mathcal{B}$ measurable iif $f$ is continuous?

Let $f$ be a function from $(\mathbb{R}, \mathcal{F}) \rightarrow (\mathbb{R}, \mathcal{B})$, where $\mathcal{F}$ is a sigma-algebra and $\mathcal{B}$ denotes the Borel sigma-algebra. Does ...
2
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1answer
21 views

Are the limits of a.e. equal sequences of measurable functions equal a.e.?

I haven't seen the following fact in any textbook or reference, which either means that it is trivial, or that it's false. Hopefully it is the former. I've attempted a proof: Claim: Let $f_n, g_n : ...
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1answer
30 views

Proving $f_n\rightarrow f$ such that $\sup_n \| f_n \|_1 \leq K$ implies $\| f \|_1\leq K$

Looking back at my notes from class, I see: Claim. $f_n\rightarrow f$ such that $\sup_n \| f_n \|_1 \leq K$ implies $\| f \|_1\leq K$. It appears after the statement and proof of Fatou's lemma but I ...