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

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Hardy-Littlewood maximal function example

Let $$f(x)= \begin{cases} \frac{1}{xlog^2x}, \hspace{2mm} \text{if} \hspace{2mm} 0 < x < \frac{1}{2}\\ 0, \hspace{2mm} \text{otherwise} \end{cases} $$ I have so far shown that $f$ is ...
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Proof about independent random variables

Let $X_1,X_2,...$ be independent random variables with $P(X_n=1)=p_n$ and $P(X_n=0)=1-p_n$ Show that $X_n\rightarrow 0$ in probability if and only if $p_n\rightarrow0$, $X_n\rightarrow 0$ almost ...
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Relations between measure of the image set of a function between manifolds and its rank

Given $M^m$, $N^n$ smooth manifolds with $\dim M=m > \dim N =n$. $f\colon M \to N$ $C^1$ of rank $k < n$. Prove that $f(M)$ is a null set. my attempt We can't use Sard Theorem, because the ...
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1answer
20 views

dominated convergence theorem and solve an integral

I will solve this integral from dominated convergence theorem can you show me how: $$\int^1_0\frac{t^{a-1}}{1+t^{b}}dt=\sum_{n=0}^{\infty}\frac{(-1)^{n}}{nb+a}$$
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2answers
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Why is convergence w.r.t $\mathcal L^p$-norm of a sequence $(f_n)$ of $\mathcal E-\mathcal B(\mathbb R)$-functions called “convergence in $p$-mean”?

In measure-theory, why is convergence with respect to the $\mathcal L^p$-norm of a sequence $(f_n)_{n \in \mathbb N}$ of $\mathcal E-\mathcal B(\mathbb R)$-measurable functions called "convergence in ...
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0answers
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Lebesgue outer measure of linear subspace

Prove that $\lambda^{*}(A)=0$, where $\lambda^{*}$ is $n$-dimensional Lebesgue measure on $\mathbb{R}^n$ and $A$ is $k$-dimensional subspace of $\mathbb{R}^n$ and $k<n$. I've proved this for ...
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1answer
9 views

Finite parameter integral implies finite norm

Need a bit of help with a parameter integral problem. We have, $X$ is a finite measure space with measure $\mu$ and $f:X\rightarrow [0 , \infty)$ is a measurable function. The parameter integral ...
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1answer
18 views

Introduction to Measure-theoretic Probability George Roussas. example 4 page 1 [on hold]

I am reading Introduction to measure-theoretic Probability George Roussas. example 4 page 1 says: Let $\Omega$ be infinite (countably or not) and let $\mathcal{C}= \lbrace A \subseteq \Omega;A$ is ...
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25 views

Give an example of an open bounded set whose boundary has no (Lebesgue, Jordan) measure zero

Calculus (believeitornot!) homework due Tuesday. They already give me a hint: consider an open cover of intervals in Q∩[0,1] whose "lengths sum up shortly!".
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19 views

Which projection, in $L_\infty$ norm or $L_2$ norm, is non-expansion?

I am just wondering which projection is non-expansion? Is there any reference to find this out.
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Orthogonal under a new inner product

Assume $C$ is a compact group of $Gl(d,\mathbb{R})$ and $\lambda$ is its normalised Haar measure. Then define a new inner product $$[x,y]=\int_C<Mx,My>d\lambda(M).$$ I was told that each element ...
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1answer
18 views

Proving functions are in $L_1(\mu)$.

Let $(\mathcal{X}, \mathcal{A}, \mu)$ be a measure space. Take $f,g \in L^1(\mu)$. Prove that $\sqrt{f^2+g^2}$ and $\sqrt{\vert fg\vert}$ are in $L^1(\mu)$. First, I prove that $h = ...
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1answer
11 views

integral over limsup of numerical function

I have to show that inquality but I donot know how? let $(\Omega,A,\mu)$ be a measure space and let $f_n$ ,n$\in$ N be nonnegative measurable numerical function on $\Omega$ ,show that $$ ...
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2answers
38 views

How to finish this proof about $\sigma$-algebras?

Let $\Omega$ be a countable set and $\Sigma$ be a $\sigma$-algebra on $\Omega$. Prove that there exists some countable partition of $\Omega$ that spans $\Sigma$. That is, prove the ...
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2answers
22 views

if $f \ge 0$ and $\int fd\mu<\infty$, then for any $a>0$ the set $\{f\ge a\}$ has finite $\mu$-measure

Let $(X,\Sigma , \mu)$ be a measure space. Show that if $f \ge 0$ and $\int fd\mu<\infty$, then for any $a>0$ the set $E_a:=\{f\ge a\}$ has finite $\mu$-measure. My attempt: We know that ...
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1answer
23 views

How to prove that the sequence of random variables converges to a random variable?

If $Z_1,Z_2,\cdots,Z_n$ are random variables such that $Z(\omega)=\lim_{n \to \infty}Z_n(\omega)$ exists $\forall \omega \in \Omega$, then $Z$ is also a random variable. I was reading a book on ...
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1answer
33 views

Measurability of $\{(x,y): x\in M,0\leq y\leq f(x)\}$

Let $(X,\mathfrak{S}_x,\mu_x)$ be a measure space endowed with the $\sigma$-additive and complete measure $\mu_x$ defined on the $\sigma$-algebra $\mathfrak{S}_x$, let $\mu_y$ be the linear Lebesgue ...
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26 views

Borel Measures: Atoms (Summary)

Disclaimer: This is a summary of the discussions: Measure Atoms: Definition? Borel Measures: Discrete & Continuous? Borel Measures: Atoms vs. Point Masses Reference: Further results are ...
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What is the definiton for “best probability measure”?

I'm looking for this definition is notes that use the phrase and elsewhere, but it just isn't there. Does anyone else recognize the phrase?
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1answer
35 views

Gaussian distributions - a question about convergence

Let $\mu_n$ be Gaussian distributions with mean $0$ and standard deviation $1/n$ and $f$ a function. It may be true that if $\underset{\mathbb{R}}{\int} f \mu_n dx \rightarrow ...
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27 views

Why this two dynamical Systems are not isomorphic?

Given two dynamical Systems on [0,1) with the Borel $\sigma-Algebra$ and the lebegue measure l. $T_a (x) = x + a$ mod1 $T_2 (x) = 2x$ mod1. Show that this two systems are not isomorphic for any ...
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1answer
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How can I say whether $g \in \bar {\mathcal M}(\mathcal B(\mathbb R))^+$ such that $\int g \ d(\lambda) < \infty$ and $|f_n| \le g$ exist?

Let $f_n(x)=\frac n 2 \cos(x)1_{[\frac {-1} n, \frac 1 n]}(x)$ be a function $\mathbb R \rightarrow \mathbb R$. I've shown that $f_n \in \mathcal L^1(\lambda)$ and $\int f_n \ d(\lambda) = n ...
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$\mu_x\otimes\mu_y$ and $\mu_y\otimes\mu_x$

Let $X$ and $Y$ spaces endowed with measures $\mu_x$ and $\mu_y$ defined on set semirings $\mathfrak{S}_x$ and $\mathfrak{S}_y$ and let $A\subset X\times Y$ be a subset of $X\times Y$ such that ...
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1answer
71 views

Consistency strength of 0-1 valued Borel measures

The following is an overly fancy way of asking a question suggested in Borel Measures: Atoms vs. Point Masses Let $\phi$ be a property that topological spaces can have (such as "compact", "$T_1$", ...
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1answer
14 views

On Egorove's Theorem

In the book Measure and Intrgral - Wheeden, Zygmund (p.57), I saw the Egorove's theorem and its proof. I puzzled with the statements of the Egorove's theorem, and a Lemma needed in the proof of ...
2
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1answer
47 views

If functions converge a.e. and their integrals converge, does convergence in $L^1$ follow?

I was wondering if $f_n, f:\mathbb{R}\rightarrow\mathbb{R}$ are s.t. $f_n\rightarrow f$ pointwise a.e. and $\int f_n\rightarrow \int f$ where integrals are Lebesgue Integrals, is there any Theorem or ...
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2answers
41 views

Question about sigma-algebras

Assume some random variables $$X_1,\dots,X_n : \Omega \to \mathbb{R}$$ are given where $(\Omega,\Sigma)$ and $(\mathbb{R},\mathcal{B}(\mathbb{R}))$ denote measurable spaces. How can one proof that $$ ...
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approximation simple functions with finite support

Let $f$ be a nonnegative measurable function. I want to prove that there is an increasing sequence of nonnegative simple functions each of which vanishes outside a set of finite measure such that ...
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1answer
9 views

Measure of a Set in Relation with a Bounded Function

If $f:[0,1]\to\mathbb{R}$ is bounded, then, for a given $\epsilon$, can the set $S:=\{x\in[0,1]:f(x)>\sup f-\epsilon\}$ be of (Lebesgue) measure $0$ ? If $f$ is continuous then I think the answer ...
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1answer
51 views

In frequentism, does every event have a probability?

For an infinitely repeatable trial with event space $\Omega$, and an event $A\subseteq \Omega$, the frequentist probability of $A$ is defined: $P(A):= \lim_{n\rightarrow\infty} \frac{n_a}{n}$, where ...
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1answer
21 views

What does $\mathbb{R}$-invariant mean for a measure?

Let $(X, A, m)$ be a measure space with m being $\mathbb{R}$-invariant. What does this mean?
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2answers
28 views

Gamma function in $C^{2}$

How can I show that for $x>0$, the Gamma function is at least $C^{2}$? The Gamma function is defined as $$\displaystyle \int^\infty_0 e^{-t}t^{x-1}\ dt$$ For which $x$ is the integrand integrable?
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2answers
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Lebesgue Measure: No Atoms!

Disclaimer: This is just meant as record of a proof. For more details see: Answer own Question How to prove that the Lebesgue measure has no atoms: $$\lambda:\mathbb{R}^n\to\mathbb{R}_+$$ (Recall ...
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Discrete set that is compact and jordan measurable

We define a) C(0) = [0,1] b) C(n) = New set that is obtained by erasing 1/3^n section long from the middle of the remaining section in C(n-1) *If C(0) 0---------1 Then C(1) = 0---xxx---1 C(2) = ...
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Cofinite Topology: Borel Algebra?

Given the cofinite topology: $$\mathcal{T}:=\{U\subseteq\Omega:\#U^c<\infty\}$$ and generate its Borel algebra: $$\sigma(\mathcal{T})=\{E\subseteq\Omega:\#E\leq\aleph_0\lor\#E^c\leq\aleph_0\}$$ Why ...
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Borel Measures: Atoms vs. Point Masses

Let a measure be $\mu:\Sigma\to\mathbb{R}_+$. Call a measurable $A\in\Sigma$ an atom if: $$\mu(A)>0:\quad\mu(E)<\mu(A)\implies\mu(E)=0\quad(E\subseteq A)$$ and a singleton $\{a\}\in\Sigma$ a ...
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Set $E$ which halves the measure of an open interval [duplicate]

This was an exam question. I know that my answer is wrong, but I believe myself to be on the right track. Can someone help me finish my construction? Here is the question. Find a set $E$ with the ...
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1answer
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If $U \cap \iota(\Bbb{R}^k)$ is a $k$-dimensional null-set for every **linear** embedding $\iota : \Bbb{R}^k \to \Bbb{R}^n$, $U$ has measure zero

In the post Does $\mu_k(U \cap \mathbb{R}^k)=0$ for all affine embeddings of $\mathbb{R}^k$ in $\mathbb{R}^n$ imply $\mu_n(U)=0$?, the OP originally posed (essentially) the following question: ...
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1answer
24 views

Show that sigma algebra generated by subsets of R contains sigma borel algebra of R.

So basically I want to show that if (a,b) is in F, then S(F) contains B(R). (b>a) (Where S(F) is the sigma algebra generated by F and B(R) is the Borel sigma algebra generated by the real numbers.) ...
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1answer
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Proof of Outer Regularity of Lebesgue Measure on R

Let $E \subseteq \mathbb{R}$ be a measurable set, and $\epsilon > 0 $. Show that there is an open set $G \supseteq E$ such that $\mu(G \setminus E) < \epsilon$. Any hints? By the definition of ...
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Equivalent Measures via Hahn-Kolmogorov and $\sigma-$finiteness

Hello Mathematics community. I am currently struggling with the following problem from Terry Tao's Introduction to Measure Theory textbook. It deals with pre-measures and the Hahn-Kolmogorov ...
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Does Fubini's theorem imply $\int_X (\int _Y f _x d \lambda )d \mu=\int _X d \mu \int _Yf(x,y) d \lambda$?

I need some help with intepretating the result of Fubini's theorem. define $ \phi (x) =\int_Y f _x d \lambda $ and $ \psi (y)= \int _X f _y d \mu$ According to Rudin, Fubinis theorem tells us that ...
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1answer
28 views

If $\mu(B)=0$ then $\mu_y(B_x)=0$

Let $B_x$ be the $x$-section of a $\mu_x\otimes \mu_y$-measurable set $B$, where $\mu_x\otimes \mu_y$, which I will call $\mu$, is the Lebesgue extension of the product measure $\mu_x\times \mu_y$ ...
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1answer
30 views

Is a function with everywhere discontinuities of the second kind always measurable?

Let $f : [0,1] \to \left\{ 0, 1 \right\}$ be a function that has at each point a discontinuity of the second kind. Is $f$ measurable if we equip the domain with the Borel or even Lebesgue ...
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Question about Girsanov theorem

Tn the book "Stochastic Differential Equation" from Oksendal one can find the following theorem(6th edition, Theorem 8.6.8): Let $X(t)=X^x(t)$ and $Y(t)=Y^x(t)$ be an Itô diffusion and an Itô ...
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1answer
18 views

Union of $x$-sections measurable?

I know that the $y$-section $A_x$ of a $\mu_x\otimes \mu_y$-measurable set $A$, where $\mu_x\otimes \mu_y$ is the Lebesgue extension of the product measure $\mu_x\times \mu_y$ (both measures being ...
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1answer
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“Lebesgue measure” on metric spaces?

Sry if my question is stupid, but I just wondered if is there is like a corresponding counterpart to the Lebesgue measure on $\mathbb{R}^n$ for (some?) metric spaces $(E,d)$? Since the natural way to ...
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Proof for the absolute continuity with respect to Lebesgue measure

Let $\mu$ be a measure. I'm looking for a reference to a proof showing that the following condition is enough to prove absolute continuity with respect to the Lebesgue measure: $$\liminf\limits_{r ...
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I know by Fubinis theorem that $N$ is $\mathcal E$-measurable and $\mu(N)=0$. How can I see that $N \neq X$, that is $N \subset X$?

Let $(X, \mathcal E, \mu)$ and $(Y, \mathcal F, \nu)$ be $\sigma$-finite measure-spaces and consider the product-space $(X \times Y, \mathcal E \oplus \mathcal F, \mu \oplus\nu)$. Let $f: X \times Y ...
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1answer
28 views

Monotone Sequence of Sets

If someone can check my proof for the following statement, would be awesome. Thanks. Suppose $\{A_n\}$ is a monotone sequence of subsets. If $A_n \downarrow$, then $\lim_{n \rightarrow \infty} A_n = ...