For questions about the Lebesgue measure, a measure defined on the Borel or Lebesgue subsets of the real line or $\mathbb R^d$ for some integer $d$. Use it with (tag: measure-theory) tag and (if necessary) with (tag:lebesgue-integral).

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Prove that set of measure zero cannot have measure theoretic density away from 1

I've been trying to solve the following question from Bass: Fix $\epsilon \in (0,1).$ If a Lebesgue measurable set $A$ is such that for every bounded interval $I$, $m(A \cap I) \leq (1-\epsilon) ...
2
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29 views

Set $E\subset \mathbb{R}^n$ of positive Lebesgue measure such that the Lebesgue measure of $\bar{E}\backslash Int(E)$ is zero

Let $E\subset \mathbb{R}^n$ have positive Lebesgue measure. What are easily interpretable sufficient conditions on $E$ to guarantee that the difference between the closure $\bar{E}$ and the interior ...
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1answer
59 views

A Vitali set is non-measurable, direct proof, without using countable additivity

I am teaching a measure theory class, where we are in the process of constructing Lebesgue measure on $\mathbb{R}$ via the usual Caratheodory outer measure construction. As motivation, we began by ...
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1answer
30 views

Characteristic function of a measurable set.

Let $X=L^p[0,1]$ $(1\leq p<\infty)$ be the Lebesgue space of p-integrable real functions on $[0,1]$. Let $D\subseteq [0,1]$ be measurable subset. The characteristic functions $\chi_D$ is defined as ...
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1answer
40 views

Show that the image of Lipschitz function $\gamma : [0,1] \to R^n$ has measure $0$, if $n \ge 2$.

Problem Statement: Let $\Gamma$ be the image of a Lipschitz continuous function $\gamma : [0,1] \to R^n$, that is, $\Gamma = \{\gamma(t) : t \in [0,1]\}$, and $|\gamma(t_1) - \gamma(t_2)| \le K |t_1 - ...
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34 views

Four definitions for Borel algebra in $\mathbb{R}$? [on hold]

Let us take $X=\mathbb{R}$, the set of real numbers. Of course we know a Borel algebra in $\mathbb{R}$. How can we have four definitions for a Borel algebra in $\mathbb{R}$?
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2answers
19 views

Subset of null set (set of measure zero)

Is there a proper subset of a set of measure zero that is not measurable? Any examples? Thanks a lot! I suspect the answer is yes due to some careful phrasing in books, e.g. let F be a subset of a ...
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24 views

Looking for a bounded set in a set with finite measure lebesgue.

Let $A\subseteq\mathbb{R}^{n}$ with $\mu^{*}\left(A\right)<\infty$. Show that for each $\varepsilon >0$ there is $A_{\varepsilon}\subseteq\mathbb{R}^{n}$ bounded such that ...
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1answer
23 views

Can this be proved using the MCT instead of the DCT?

I've seen various version of the DCT prove that if $f$ is a real valued, or extended real valued, or complex, integrable function, and if $\{E_n\}_n$ is a sequence of disjoint measurable subsets, ...
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0answers
21 views

$\mu$ is a finite Borel measure on $\Bbb R$, absolutely continuous w.r.t. to the Lebesgue measure $m$. Prove that $x \mapsto \mu(A+x)$ is continuous.

Let $\mu$ be a finite Borel measure on $\Bbb R$, which is absolutely continuous with respect to the Lebesgue measure $m$. Prove that $x \mapsto \mu(A+x)$ is continuous for every Borel set $A \subseteq ...
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23 views

Does a proper compact subset of a compact subset of R has strictly smaller measure?

Let K be a compact subset of R and H a proper compact subset of K. Does H has a strictly smaller Lebesgue measure than K?
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1answer
63 views

Use DCT to show:

$\lim_{n \rightarrow \infty} \int_0^{\infty}f_n(x)dx = \int_0^{\infty} \frac{x}{e^x-1}dx$, where $f_n(x):=\frac{n}{e^x-1}\sin\frac{x}{n}$ Hi I'm working on some practice questions and having a bit of ...
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2answers
53 views

If $\mathcal{B}$ is a base of a topology space $\left(X,\tau\right)$, then the Borel $\sigma$-algebra is generated by $\mathcal{B}$?

Let $\left(X,\tau\right)$ a topology space and $\mathcal{B}$ a base of the topology, my question is: The Borel $\sigma$-algebra is generated by $\mathcal{B}$ ?
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0answers
16 views

Application of Carathéodory outer measure theorem

We know Carathéodory outer measure theorem and its proof, but I want an application or example about this theorem (I mean give me an outer measure and show the class $Μ$) I don't want this outer ...
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0answers
48 views

Differentiability of parameter-dependent integrals when derivative exists only almost everywhere

This unanswered question asked in 2013 Differentiation under the Integral Sign (let's call this Q-zero) seems to be taken from this (or pdf ver.). The result on differentiation under the integral ...
2
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1answer
36 views

Is the outer measure of $[0, 1] $ equal to $0$?

I am going to try and prove that the outer measure of $[0, 1] $ is $0$. I would be grateful if someone could point out the mistake. The outer measure of an interval is defined as $\inf \Sigma {l ...
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0answers
16 views

The dual of the space of $p$-locally integrable functions

If $X$ is a space of finite measure, what is the dual space of $L^p _{loc}$ (the space of locally $p$-integrable functions)? When $p=1$, a good answer has already been provided. What is known for $p ...
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0answers
38 views

Exercise 3.32 from Real Analysis of Folland

Can someone give me some hint on how to solve this problem? Thanks a lot If $F_1, F_2, ..., F \in NBV$ and $F_j \rightarrow F$ pointwise, then $T_F \le \liminf T_{F_j}$ Here, NBV is the ...
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0answers
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Is there an easy proof that the set of $x \in [0,1]$ whose limit of proportion of 1's in binary expansion of $x$ does not exist has measure zero?

So for given $x \in [0,1]$, if we let $f_n(x)$ be the fraction of 1's occurring in the first $n$ binary digits of the binary expansion for $x$ (where we always assume an infinite trailing string of 0 ...
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3answers
38 views

Is the intersection of the following closed and open set closed? Generally?

Ok, I have been informed that the below lemma is incorrect. I needed it to prove the following statement. Could someone else provide a proof? Statement: If m(E) is finite, there exists a compact set ...
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4answers
135 views

Can we find uncountably many disjoint dense measurable uncountable subsets of $[0,1]$?

Can we find uncountably many disjoint dense measurable uncountable subsets of $[0,1]$? Obviously we may as well assume all the subsets have measure $0$. If I didn't specify the subsets were ...
2
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1answer
23 views

Sets cut into two halves of equal size by any straight line through a particular point

Is there an easy characterization of all sets $M \subseteq \mathbb{R}^2$ with the following property? A point $(x_M,y_M)$ (which may depend on $M$) exists such that each straight line through ...
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1answer
120 views

Is $\overline{D}_{\varepsilon}$ a connected Jordan region in $\mathbb{R}^{n}?$

Definition. Let $E$ be a nonempty subset of $\mathbb{R}^{n}$.The distance from a point $\mathbb{x}\in\mathbb{R}^{n}$ to set $E$ is defined by ...
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1answer
66 views

Property of a set of a positive Lebesgue measure

I am trying to see whether it is true that in any set of a positive Lebesgue measure in $R^2$ we can always find two points $(a_1,a_2)$ and $(b_1,b_2)$ such that the following hold: $a_1>b_1$ ...
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1answer
37 views

Repeated extension of Lebesgue measure

In Halmos' Measure Theory, section 16, exercise 2 deals with the extension of a $\sigma$-finite measure $\mu$ defined on a $\sigma$-ring $S$ to any set $M$ in the hereditary $\sigma$-ring induced by ...
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1answer
36 views

L'Hôpital with absolute continuity

I have been studying for my real analysis qualifying exam, and I have noticed a trend of questions similar to the following: Suppose that $f$ is absolutely continuous, $f'\in L^3$, and $f(0)=0$. ...
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1answer
35 views

Properties of decreasing sequence of Lebesgue measurable sets.

I'm trying to prove a property of Lebesgue measure sets that says: If the $A_{k}$'s are measurable and $A_{1} \supset A_{2} \supset A_{3} \supset \ldots,$ and if $\lambda (A_{1}) < \infty, $ then ...
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1answer
42 views

Lebsegue measure of $\{ 0<x \leq 1: x \sin \left(\frac{\pi}{2x}\right) \geq 0 \}$

Find the Lebsegue measure of the set $A= \left\{ 0<x \leq 1: x \sin \left(\frac{\pi}{2x}\right) \geq 0 \right\}$. The answer given is $1 - \ln \sqrt{2}$. My thought: I only know that Lebsegue ...
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0answers
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$f$ and $f'$ are in $L^1 (\Bbb R)$. Prove that $\int_{-\infty}^{\infty} f' (x)dx=0$. [duplicate]

Problem: Suppose $f: \Bbb R \rightarrow \Bbb R$ is absolutely continuous on every interval $[a,b]$, and that both $f$ and $f'$ are in $L^1 (\Bbb R)$. Prove that $\int_{-\infty}^{\infty} f' (x)dx=0$. ...
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0answers
24 views

Find the limit $\lim \int_{(0,1]} f_n \, d\lambda $ where $ \lambda $ is lebesgue measure

Find the limit $\lim \int_{(0,1]} f_n \, d\lambda $ where $ \lambda $ is lebesgue measure and $ f_n(x)=\dfrac{|\cos(x^{-2})|}{x^{1-1/n}} $ for $ x\in (0,1] $ Is there lebesgue integrable function $g$ ...
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1answer
16 views

existence of a sequence of continuous functions with two conditions

$\displaystyle \int_0^1 \lim_{n\to\infty} f_n(x)\,dx = \lim_{n\to\infty}\int_0^1f_n(x)\,dx $ There is no function $\,g:\left[0,1\right]\to \mathbb R\,$ lebesgue integrable such that $\,\left\lvert ...
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1answer
28 views

Borel measure induced by the Cantor function?

In an example to measure being mutally singular, the book has an example I do not understand. First the book has the definition: Mutually Singular Measure Let $(\Omega,\mathcal{A})$ be a ...
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1answer
33 views

Let $([0,1],\mathcal{B}([0,1]),\lambda)$, $\lambda$ Lebesgue measure in $[0,1]$.

Show that if $f$ is $p$-integrable then, for each $\epsilon>0$, exists a function $h$ which is continuous in $[0,1]$ s.t. $\|f-h\|_p\leq\epsilon$. Is there any simpler way to show it than ...
2
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2answers
64 views

$\mu(A \cap I) \le a \mu(I)$ implies $\mu(A) = 0$?

Let $\mu$ be lebesgue measure on $\mathbb{R}$, $0<a<1$. If $\mu(A \cap I) \le a \mu(I)$ holds for any interval $I$, can I say $\mu(A)=0$? I tried to construct a counterexample by considering ...
2
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1answer
30 views

Showing a certain function vanishes almost everywhere

Can someone give me a hint on the following problem? I'm not sure what to do... Suppose $f\in L^1([0,1])$ is such that for all $n=0,1,2,...$ we have $$\int_0^1 f(x)(\sin x)^n\,dx = 0.$$ Show that ...
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54 views

Is the completion of a measure space necessary?

Most important theorems in measure theory do not assume the completeness of measure spaces. Monotone convergence theorem, Dominated convergence theorem, and Fubini's theorem, to name a few. So I ...
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2answers
94 views

Number of equivalence classes of functions of real variable with the a.e relation.

What is the cardinal of the set $\mathcal{F}(\mathbb{R};X)/ \sim$ where $\sim$ is the relation $f\sim g \iff \mu(\{x\in \mathbb{R};f(x)\ne g(x)\})=0$ and $|X|=|\mathbb{R}|$? I guess that is ...
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1answer
43 views

if $\mu(X)$ is finite and $f$ is finite on X a.e then $\lim_{n\to \infty}\mu \{x: |f(x)|\geq n\}=0$

Let $(X,\mathcal F, \mu)$ be measurable space with $\mu(X)<\infty$. Prove that if function $f$ is measurable and finite on $X$ then $$\lim_{n\to \infty}\mu \{x: |f(x)|\geq n\}=0.$$ I have been ...
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1answer
38 views

Mean value theorem for Lebesgue integral

Let $f$ be a mesurable function and $g$ be integrable function, and $\alpha, \beta$ are real numbers such that $\alpha \leq f \leq \beta$ a.e . Prove that there exists a real number $\gamma \in ...
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3answers
33 views

Clarification on the definition of Lebesgue measure

I'm doing some independent reading on the Lebesgue measure and I have the following questions: 1) on the definition of a Lebesgue measurable set: Let $E \subseteq \mathbb{R} $. On Wikipedia, $$\mu(E) ...
5
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2answers
49 views

Is $|\bar{A}| = 0$ if every point of $A \subset \Bbb{R}$ is isolated from the right?

Consider a subset $A$ of $\Bbb{R}$ such that every points in $A$ is isolated from the right in the following sense $$ \forall a \in A,\ \exists \epsilon > 0 \ \text{ s.t. } \ A \cap [a, ...
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2answers
51 views

What's the relationship between a measure space and a metric space?

Definition of Measurable Space: An ordered pair $(\Omega, \mathcal{F})$ is a measurable space if $\mathcal{F}$ is a $\sigma$-algebra on $\Omega$. Definition of Measure: Let $(\Omega, F)$ ...
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1answer
35 views

If $f, g$ are measurable functions, then $f+g$ is measurable

Show that $f(x)+g(x)<a$ iff there exists rational number $r,q$ such that $r+q<a$ and $f(x)<r; g(x)<q$. Use this to prove if $f, g$ are measurable functions, then $f+g$ is ...
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2answers
31 views

if $\mu(X)$ is finite then $\lim_{n\to \infty}\mu \{x: |f(x)|\geq n\}=0$

Let $(X,\mathcal F, \mu)$ be measurable space with $\mu(X)<\infty$. a) Prove that if function $f$ is measurable on $X$ then $$\lim_{n\to \infty}\mu \{x: |f(x)|\geq n\}=0.$$ b) Can we ...
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0answers
16 views

proof coordinate functions of integrable function $f: \mathbb{R}^n \rightarrow \mathbb{R}$ integrable

If $$f(x)=f_1(x_1)\cdots f_n(x_n)$$ and $f$ is an integrable function from $\mathbb{R}^n$ to $\mathbb{R}$. Proof that $f_i(x_i)$, $i = 1, \ldots, n$ are integrable. With the Fubini theorem?
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1answer
42 views

What does Lebesgue measure space look like?

Definition of Measurable Space: An ordered pair $(\Omega, \mathcal{F})$ is a measurable space if $\mathcal{F}$ is a $\sigma$-algebra on $\Omega$. Definition of Measure: Let $(\Omega, F)$ ...
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2answers
72 views

Null set squared is a null set

I'm attempting to find a solution to the following problem that doesn't involve splitting this into various cases. The question is: "If $m^*(E) =0$, show that $m^*(E^2) = 0$, where $E^2 = \{x^2 ...
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1answer
21 views

prove that $\lim_{m \rightarrow \infty} \Sigma_{k=-m^2}^{m^2}|\int^{(k+1)/m}_{k/m}f(x)dx|=\|f\|_{L^1 (\Bbb R)}$.

Suppose $f \in L^1 (\Bbb R)$, prove that $$\lim_{m \rightarrow \infty} \sum_{k=-m^2}^{m^2}\left|\int^{(k+1)/m}_{k/m}f(x)\,dx\right|=\|f\|_{L^1 (\Bbb R)}.$$ For this one, it's easy to prove when $f$ ...
2
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0answers
27 views

Caratheodory criterion, sufficient but is it necessary?

When constructing the Lebesgue measure on $\mathbb{R}$, it is shown that the sets that satisfy the caratheodory criterion form a sigma-algebra, and also that the countable additivity of the Lebesgue ...
0
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1answer
42 views

A problem about Hardy-Littlewood maximal function from Folland real analysis book

For $x \in \Bbb R^n$, define $H^* f(x)=\sup{\frac{1}{m(B)}\int_B|f(y)|\,dy}$, where $B$ is a ball and $x \in B$ and $$H f(x)=\sup_{r>0} \frac{1}{m(B(x,r))} \int_{B(x,r)}|f(y)| \, dy.$$ Show that ...