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

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391 views

Lebesgue integral of a bounded measurable function over a measurable subset of a measurable set of finite measure. [duplicate]

Let $f$ be a bounded measurable function on a set $E$ of finite measure. For a measurable subset $A$ of $E$, show that $\int_A f=\int_E f\cdot\chi_A$, where $\chi_A$ is a characteristic function on ...
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3answers
68 views

Simple Function attains a maximum?

Let $A_1,..,A_n \subset \mathbb{R}^n$ be sets with finite measure and let $a_1,\ldots,a_n$ be real numbers. Consider the simple function $$f(x)=\sum_{k=1}^n a_k \chi_{A_k}$$ where $\chi_A$ is the ...
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0answers
82 views

Change of differentation and integration signs.

I'm going through an old exam in a course I'm taking. I have the given rule: Let $X$ be a measure space, $U$ be open subset in $\mathbf{C}$ and $f: U \times X \to \mathbf{C} $ be a function s.t. the ...
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1answer
390 views

Markov/Chebyshev Inequality

I am looking at proofs of Markov or Chebyshev's inequality that for a measurable function, the set $B=\{x\in\mathbb R^n:|f(x)|\ge t\}$ where $0\lt t\lt \infty$ , has a measure that is ...
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0answers
35 views

When a family of measures provide continuity?

Consider a mapping $m: \mathcal{B}(X) \times P \rightarrow [0,1]$, where $X \subseteq \mathbb{R}^n$, $P \subseteq \mathbb{R}^m$, and $\mathcal{B}(X)$ denotes the Borel sets. $\forall p \in P$, ...
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2answers
90 views

Does $f_n \to 0$, a.e., implies $\int_{\mathbb R} \sin(f_n(x)) dx \to 0$, when each $f_n \in L^1$

Let $\{f_n\}$ be a sequence of $L^1(\mathbb R)$ functions converging a.e. to zero. Does $$ \lim_{n\to \infty} \int_{\mathbb R} \sin(f_n(x)) dx = 0? $$ I think the answer is no, but I can't find a ...
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1answer
66 views

Prove there is a Borel measure u such that $u[x,y) = a(y) - a(x)$

If anyone has a solution to the following exercise, I would be grateful. Thanks. Let $\alpha$ be continuous and increasing on $[a,b]$. Prove that there exists a unique Borel measure $\mu$ on ...
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1answer
61 views

Prove a function $f_\sigma\in C^\infty(\mathbb{R})$

Define $C^\infty(\mathbb{R})$ be the space that contains all bounded continuous functions on $\mathbb{R}$ that has continuous derivatives of all orders. Suppose $K\in C^\infty(\mathbb{R})$ s.t. ...
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3answers
86 views

A condition that balls have finite measure

Let $(X,d)$ be a metric space and let $\mu$ be a positive measure on $X$. I want to require that $(X,d)$ and $\mu$ have either of the following properties: $\forall y \in X$, $\forall r \geq 0$, ...
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1answer
27 views

If a function is $L^p$ small, is its expectation with respect to a $\sigma$-algebra $L^p$ small?

This came up in my homework, but isn't strictly my homework. I've just gotten very curious, and I keep going in circles trying to prove it. Consider a probability measure space $(X,\Sigma,\mu)$ and ...
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1answer
313 views

Random Walk on Z

Let $S_n$ be the symmetric random walk on $\mathbb{Z}$. How do i calculate $P(\limsup_{n\rightarrow\infty} S_n=\infty)$? I already know that the probability is 1 but I don't really know how to start? ...
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1answer
57 views

Inequalities for bounded lipschitz functions

Suppose $X$ is a metric space with metric $d$. Define $\lVert f(x)\rVert_\infty = \sup_x |f(x)|$ and $\lVert f(x)\rVert_{LIP}=\sup\{\frac{|f(x)-f(y)|}{d(x,y}:x\not=y\}$, let $\lVert ...
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2answers
224 views

Interchanging closed operators and integrals

I am dealing with a problem in Evans PDE without measure theory knowledge... We have contraction semigroup $\{S_t\}_{t \geq 0}$ on real Banach space $X$, i.e family of bounded linear operators from $ ...
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1answer
206 views

Smallest $\sigma$-algebra generated by $\mathcal C$?

Can someone explain how this $\sigma$-algebra is attained? It's mainly the $X\cup Y$ bit which I don't understand. Question: If $\Omega = \{1, 2, 3, 4\}$ and we have a collection of sets $\mathcal C ...
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2answers
165 views

Part of proof 11.10 in Rudin's Principles of Mathematical Analysis

There is a part of proof 11.10 that I don't get in Rudin's Principles of Mathematical Analysis (3rd edition). The whole theorem is the following two statements: $\mathcal{M}\left(\mu\right)$ is a ...
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2answers
45 views

Is $\frac{|C\cap B_r(x)|}{|B_r(x)|}$ decreasing in $r$?

Suppose $C$ is a measurable set, $x\in C$, is $$ \frac{|C\cap B_r(x)|}{|B_r(x)|} $$ decreasing in $r$? Or any counterexamples? Thanks! Edit: @user39992 and @Karolis Juodelė show that it can not ...
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1answer
238 views

$\sigma$-algebras and independent stochastic processes

Let $(\Omega,\mathcal{F},\mathbb{P})$ be a complete probability space. We consider a Wiener process $W$ with respect to his standard filtration $(\mathcal{F}_t^W)_{t \geq 0}$ and a process $X$ with ...
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0answers
82 views

Riesz Representation Theorem and Indicator Function

I've been dealing with the Riesz Representation Theorem for measures and it is obvious that having a measure $\mu$ I can get a continuous linear functional $\mu^*$ in $C(X)^*$ where $X$ is a compact ...
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1answer
82 views

Why does this inequality for all characteristic functions imply it for simple functions?

This question is probably obvious, but I'm not seeing how to obtain it. A simple function is said to be finitely simple if its support is of finite measure. Let $(X_1,\mu_1)$, $(X_2,\mu_2)$, and ...
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1answer
48 views

$L^p$ convergence proof check

I don't have much experience with measure theory, so I want to make sure that I'm not making any bad mistakes. I also want to be sure that the theorem is true so I can use it. Theorem: Let $\{u_i\}$ ...
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1answer
85 views

If $B\times \{0\}$ is a Borel set in the plane, then $B$ is a Borel set in $\mathbb{R}$.

I'm trying to figure out how to prove the following "obvious" fact: Let $B\times \{0\}\subset \mathbb{R}^2$ be a Borel set, then $B\subset \mathbb{R}$ is a Borel set. The problem here is that I ...
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1answer
214 views

A question about the proof of Rademacher theorem

I'm referring to the proof of Rademacher theorem due to C.B.Morrey (i'm reading it on Simon: 'Lectures on geometric measure theory').\ The proof can be summarized in the following steps:\ 1)For every ...
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2answers
776 views

Conditional expectation on more than one sigma-algebra

I'm facing the following issue. Let $X$ be an integrable random variable on the probability space $(\Omega,\mathcal{F},\mathbb{P})$ and $\mathcal{G},\mathcal{H} \subseteq \mathcal{F}$ be two ...
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2answers
338 views

Product sigma algebra of Borel sigma algebra and Power set.

Let $([0,1],\mathcal{B},m)$ be the Borel sigma algebra with lebesgue measure and $([0,1],\mathcal{P},\mu)$ be the power set with counting measure. Consider the product $\sigma$-algebra on $[0,1]^2$ ...
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1answer
75 views

$f_n\to f $ in $L^1$ $\implies$ $\sqrt{f_n}\to\sqrt{f}$ in $L^2$?

Suppose that $\{f_n\}$ is a sequence of measurable functions converging to $f$ in $L^1(\mathbb{R}^n)$. Is it true that $\sqrt{f_n}$ converges to $\sqrt{f}$ in $L^2(\mathbb{R}^n)$? If this is true ...
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1answer
52 views

Cubes covering a set in $\mathbb{R^3}$

Let's say I divided $\mathbb{R^3}$ with 3 mutually orthogonal systems of planes and the distance between two neighboring planes of each system is $\varepsilon$. So basically what I have is countable ...
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1answer
35 views

fraction of $L^\infty$ functions

Let $\Omega $ be a bounded domain of $R^n$, and let $a, b \in L^\infty(\Omega) $ such that $ : \frac{a}{b} 1_{\{x\in \Omega/ b(x)\ne 0\}} \in L^\infty(\Omega). $ Does the following implication hold ...
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1answer
201 views

Variation of a signed measure -

I am studying Measure Theory, using the Bartle's book "Elements of Integration and Lebesgue Measure" and I couldn't do the exercise 3.Q: "If $\mu$ is a charge on $X$, let $\mathcal{v}$ be defined ...
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0answers
44 views

A question about Lebesgue measure 3

Let $ L^k $ be the k-dimensional lebesgue measure. Let $ A \subset R^n $ be a Borel set. Suppose we have proved that $ L^1(A \cap l )=0 $ for each line $ l $ parallel to some line passing throught the ...
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1answer
21 views

How to show $\dim_\mathcal{H} f(F) \leq \dim_\mathcal{H} F$ for any set $F \subset \mathbb{R}$ and $f$ continuously differentiable?

Let $f: \mathbb{R} \to \mathbb{R}$ be differentiable with continuous derivative. I have to show that for all sets $F \subset \mathbb{R}$, the inequality $$\dim_\mathcal{H} f(F) \leq \dim_\mathcal{H} ...
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1answer
2k views

Lebesgue Integral but not a Riemann integral

Is it possible for a function to be a Lebesgue integral, but not a Riemann integral? After the comments below I realize my question was not a good one. Thank you. This is my edited version: Let $f$ ...
4
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3answers
953 views

Doubt in Scheffe's Lemma

While reading up on "Glivenko Cantelli Theorem" from Probability Models by K.B Athreya, the author used 2 lemmae to prove it. One was called Scheffe's lemma, the other Polya's theorem. Scheffe's ...
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1answer
208 views

Uniformly Integrable of sets in $L_{1}(\mu)$ is equivalent to almost order boundedness

A bounded set $F\subseteq L_{1}(\mu)$ is said to be uniformly integrable if : $\forall \epsilon$ there is a $\delta>0 $, such that $\forall$ measurable set $A$, and $\forall f\in F$ , if ...
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1answer
98 views

Showing existence of Borel subsets having prescribed measure

Suppose that $\mu$ is a Radon measure on $X$ such that $\mu(\{x\})=0$ for all $x\in X$, and $A$ is a Borel set satisfying $0<\mu(A)<\infty$. I am trying to show that for any $\alpha$ such that ...
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1answer
161 views

Counter example to an adaptation of the Riesz-Markov theorem.

Suppose that $(K,\tau)$ is a topological space and that $\phi$ is a positive linear functional on $C(K)$. Then is it true that there exists a unique Baire measure $\mu$ on $K$ such that $\phi(f) = ...
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1answer
211 views

Convergence of $L^p$ norm as $p \downarrow 0$ [duplicate]

Consider a measurable space $(\Omega, \mathscr{F}, P)$ with $P(\Omega) = 1$. Define for measurable functions $X$ the following $\| X \|_p := \left(\int |X|^p dP\right)^{1/p}$. We know that for $p \in ...
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0answers
47 views

All finite Baire measures are Closed-regular?

Given a finite Baire measure $\mu$ on a topological space $X$, is it true that $\mu$ is closed-regular? Where closed regular means that, $$\mu(A) = \sup\{\mu(K)| K \space \text{is a } Z\text{-set}, ...
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1answer
96 views

Uncountable sum vs Riemann Integral

Say we have $f(x)=1$ for $x\in \Re $ $\sum_{x\in [0,1]} f(x)=\infty$ but $\int_{[0,1]}f(x)dx=1$ I thought it was intuitive to think that integrals are just representations of infinite sums but I'm ...
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1answer
270 views

Analysis - Fourier Transforms - show that convolution of characteristic functions is continuous

I would appreciate any instruction on the following exercise from real and complex analysis: Suppose $A$ and $B$ are measurable subsets of $\Re^1$, having finite positive measure. Show that the ...
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1answer
582 views

Scheffe's lemma with dominated convergence

Suppose that $f_n, f$ are non-negative measurable functions with $\mu(f_n)$ and $\mu (f)$ finite and such that $f_n\to f \text{ a.e.}$, Then $\mu(|f_n-f|)\to 0 \iff \mu(f_n)\to\mu(f)$. For ...
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2answers
88 views

A question involving Invariant Set in ergodic theorem

I have a question about the invariant set in the ergodic theorem, I am wondering if anyone could give me some help or hint. In the measurable space (X, $\Sigma$) and consider a measurable self map T, ...
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1answer
44 views

Inequliaty about indefinite integral and measurable function

$f:\mathbb{R}\rightarrow\mathbb{R}$ be Lebesgue measurable, show that $$F(x)=\int_{0}^{x}f(t)dt$$ satisfies $|F(x)-F(y)|\leq C|x-y|^{1/2}$.
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2answers
86 views

$\mu(A) = 0 \;\;\; \Rightarrow \;\; \int_A f \,\, d\mu = 0$

I'm ashamed to have to ask this question... After poring over a few measure theory text books for the last couple of hours I still cannot figure out which theorem of "standard" measure theory, out of ...
3
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1answer
447 views

Prove a set is measurable

Let $n_1 < n_2 < \cdots$ be an infinite strictly increasing sequence of positive integers. Show that $\{x \in [0, 2\pi]: \{\cos(n_kx)\}_k^\infty \text{ converges}\}$ is measurable. I honestly ...
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0answers
164 views

C* algebra of bounded Borel functions

Let $T\in B(H)$ is normal, and $B(\sigma(T))$ denote the $C^*$ algebra of all bounded Borel functions on $\sigma(T)$. Then is it true that $B(\sigma(T))$ is a closed $C^*$ algebra under the sup. norm ...
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1answer
40 views

How do I prove this limit is 0?

If $f\in L^{1}(X,\mathcal M, m)$, then $$\lim_{n\rightarrow \infty} n m\{x:|f(x)|\geq n\}=0$$
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2answers
403 views

Differentiability of convolution

First let me say that I have used the search bar and looked through all the "differentiability of convolution" questions that I saw, but none of them seem to cover this case. (If one of them did and I ...
2
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0answers
28 views

Interchange product $\sigma$-algebra and countable intersection [duplicate]

If $(\mathcal F_n)_{n\in\mathbb N}$ is a decreasing sequence of $\sigma$-algebras and $\mathcal G$ another $\sigma$-algebra, is it possible to interchange the intersection with the product ...
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1answer
33 views

$\mathbb{P}(\{X>a\}) = 1 \Rightarrow \mathbb{E}(X)>a$

The implication $$\mathbb{P}(\{X>a\}) = 1 \Rightarrow \mathbb{E}(X)>a$$ seems obviously true to me, but I can't nail a half-way rigorous proof of it. (Coming up with a counterexample seems to ...
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0answers
161 views

Relationship between Lebesgue–Stieltjes measure and regular Borel measure

From http://en.wikipedia.org/wiki/Lebesgue%E2%80%93Stieltjes_integration: The Lebesgue–Stieltjes measure is a regular Borel measure, and conversely every regular Borel measure on the real line is ...