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

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Density of $L^\infty(\Omega)h$ in $L^p(\Omega)$ where $h \in L^p(\Omega)$

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
16 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
27 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
20 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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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
48 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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21 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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14 views

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

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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16 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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43 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
29 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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21 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
31 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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2answers
34 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
28 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
24 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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0answers
4 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
29 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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19 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
30 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
12 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
23 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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0answers
24 views

Expected value and distribution of a random walk (continuous time, discrete state space)

I'm having trouble with a rather simple calculation: Let $(X_t)_{t\geq0}$ be a simple random walk in continuous time on the integer grid $\mathbb Z^d$. Let $\mathbb P_x$ and $\mathbb E_x$ denote ...
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29 views

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
72 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
22 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? ...
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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 ...
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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 ...
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1answer
36 views

Continuity of $F(x)=\int_{(-\infty,x]}fd\lambda$

For a homework assignment I was told to prove that given $f\in L^1(\mathbb R)$, the following function is continuous $$F(x)=\int_{(-\infty,x]}fd\lambda.$$ I thought to use DCT and show sequential ...
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1answer
21 views

Example of a Lebesgue unmeasurable function f such that f*f is Lebesgue measurable

Giv an example of a Lebesgue unmeasurable function $f:[0,1]\rightarrow \mathbb{R}$ such that $f^2$ is Lebesgue measurable.
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1answer
34 views

Poincare' recurrence theorem in measure theory.

I want to propose a problem, it's a version of Poincare' Recurrence Theorem, it's very similar to another problem proposed in this forum, but a bit different: Another version of the Poincaré ...
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1answer
22 views

Non-Borel a.e limit of Borel functions

As a homework assignment I'm supposed to prove or disprove Borel measurability is closed under a.e convergence. I think this is not true because the Borel $\sigma$-field is not complete. However, I'm ...
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Measurable real functions from $\sigma$-algebra generated by finite partitions

I was given the following homework problem. Let $f:X\rightarrow \bar{\mathbb{R}}$ a set function and $X$ be a measurable space whose $\sigma$-algebra is generated by a finite partition $E_1,\dots ...
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1answer
23 views

uniform Distribution on uncountable Lebesgue $0$-Sets

I know that for every measurable Set A it is possible to create a uniform Distribution on A if - A is finite - A is not a lebesgue 0-set and its not possible for infinite countable sets so I ...
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2answers
33 views

Why is Monotone Convergence Theorem restricted to a nonnegative function sequence?

Monotone Convergence Theorem for general measure: Let $(X,\Sigma,\mu)$ be a measure space. Let $f_1, f_2, ...$ be a pointwise non-decreasing sequence of $[0, \infty]$-valued ...
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2answers
157 views

Are there sets of zero measure and full Hausdorff dimension?

I would like to ask the following: Are there "many" sets, say in the interval $[0,1]$, with zero Lebesgue measure but with Hausdorff dimension $1$? The motivation for this question is the ...
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2answers
34 views

If $F$ is finite then is $\sigma(F)$ also finite?

Let $F\subset X$ be a finite family of sets of $X$. Is the sigma-algebra generated by $F$ ($\sigma(F)$) also finite? I was trying to use induction: If $F$ has one element say $A$ then ...
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1answer
18 views

Representing $C(X)$ as multiplication operators on $L^p$

Suppose that $X$ is a compact Hausdorff space and I represent $C(X)$ isometrically in $B(L^p(X,\mu))$ as multiplication operators for some finite positive regular Borel measure $\mu$. If I remember ...
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57 views

Lebesgue versus Riemann integrable

Can a Lebesgue measurable function be modified on a set of first category so as become continuous except on a set of Lebesgue measure zero? OR Can a Baire-measurable function be modified on a set of ...
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1answer
52 views

Why a function in a measure space is random variable?

Let $(\Omega,\mathcal{F})$ be a measure space and $X$ mapping from $\Omega$ to $\mathbb{R}$. Assume that $X^{-1}((a,b])\in \mathcal{F}$ for all intervals. Prove that $X$ is a random variable. ...
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3answers
47 views

A footnote about outer measure

This is the theorem about in Royden's real analysis book. And in the book there is a footnote I am confusing: Can anyone help me understanding it with examples~~~
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1answer
26 views

Null set of reals

I'm having trouble to understand a step of a proof. Let $S$ be a subset of $\mathbb{R}$. Prove that $S$ is null (Lebesgue measure). The book says the following: "It is clear that we can restrict ...
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1answer
140 views

Convergence of Riemann sums of a periodic function

Short version for people who don't like reading: Let $f\colon\mathbb{R}\to\mathbb{R}$ be $1$-periodic, measurable and bounded. Is it true that, for almost all $x$, the average of $f(x)$, ...
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1answer
39 views

Weak convergence and $\lim_{n\to \infty} \|f_n\|_{L^p}=\|f\|_{L^p}$ imply norm convergence.

Consider a $\sigma$-finite measure space $(X,A,\mu)$ and $f,f_n\in L^p(\mu)$ with $1<p<\infty$. If $f_n \stackrel{w}{\to} f$ and $\lim_{n\to \infty} \|f_n\|_{L^p}=\|f\|_{L^p}$ hold, then ...