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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Definition of integrability for sequences

My text book does not provide much about counting measures and integration. So I decided to setup integration on space $(N , P(N) , \mu_c ,R)$ myself imitating the construction of Lebesgue integral. ...
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1answer
16 views

Counting measure on sigma algebra power set of natural numbers .

My text book does not provide much about counting measures and integration. So I decided to setup integration on space $(N , P(N) , \mu_c ,R)$ myself imitating the construction of Lebesgue integral. ...
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1answer
20 views

Congruent measurable sets

I have a question regarding Congruent relations: In Euclidean geometry, two subsets of $\mathbb{R}^{d}$ are said to be congruent if one set can be mapped onto the other by translations and rotations. ...
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1answer
19 views

$g(x) = sup_{α∈A} (f_α(x))$, $x ∈ E$ need not be a measurable function.

We know that if $(f_n)$ is a sequence of measurable functions on $E$, then $g = sup_n f_n$ defined as $g(x) = sup f_n(x)$, $x ∈ E_ n$ is a measurable function. Prove by an example that if $A$ is an ...
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1answer
38 views

Prove that there is no continuous function $f : \Bbb R → \Bbb R $ such that $f = χ_I$ almost everywhere on $\Bbb R$.

Let $I = [0,1]$ and $χ_I : \Bbb R → \Bbb R$ be the characteristic function on $I$. Prove that there is no continuous function $f : \Bbb R → \Bbb R $ such that $f = χ_I$ almost everywhere on $\Bbb R$. ...
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3answers
113 views

Prove $\lim_{n \rightarrow \infty} n \int_{\frac{1}{n}}^{1} \frac{\cos(x+\frac{1}{n})-\cos(x)}{x^{\frac{3}{2}}}dx$ exists.

Prove $\displaystyle\lim_{n \rightarrow \infty} n \int_{\frac{1}{n}}^{1} \frac{\cos(x+\frac{1}{n})-\cos(x)}{x^{\frac{3}{2}}}\,dx$ exists. I want to use Dominated convergence theorem to show the ...
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1answer
24 views

If $R$ be a Union of zero measure sets , what is the cardinal of index set? [duplicate]

If $R$ be a Union of zero measure (lebesgue) sets , what can we say about the cardinal of index set? Does this question related to continuum hypothesis? Thanks.
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1answer
12 views

Which measure gives this exterior mesure $\mu^*$?

Let $\mu^*:\mathcal P(\mathbb R^2)\longrightarrow \mathbb R$ defined by $$\mu^*(E)=\inf\left\{\sum_{i=1}^\infty m(T_i)\mid E\subset \bigcup_{i=1}^\infty T_i\right\}$$ where $T_i$ are triangles and $m$ ...
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18 views

Is the following construction a measure?

Please excuse my informality, but I am currently working on the intuition behind the idea. Suppose two parameters are given: An event $E$ composed of basic events, like union and intersection of ...
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14 views

Linear transformations and Lebesgue measure

Let $d,d'\in\mathbb{N}$, $d'<d$, and for all $i\in\{1,\ldots,d\}$ let $A_i\in\mathbb{R}^{(d-1)\times d}$ be the matrix one gets if he cancels the $i$-th row of the $d$-dimensional identity matrix, ...
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14 views

Construct an explicit isomorphism between standard measurable spaces

I am looking for explicit measure-preserving bijections between any pair of $\mathbb{R}$, $\mathbb{R}^n$, $\mathbb{R}^\infty$, and $\{0, 1\}^\infty$.
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14 views

Give example $\{E_n\}$ s.t $m^*(\cup E_n)<\sum m^*(E_n)$ and $m^*(\cap E_n)<\lim m^*(E_n)$.

Give an example of disjoint sequence of sets $\{E_n\}$ s.t $m^*(\cup E_n)<\sum m^*(E_n)$. Give an example of sequence of decreasing sets $\{E_n\}$ $m^*(E_i)< \infty$ s.t $m^*(\cap ...
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1answer
29 views

Understanding the use of Lebesgue integrals

I need some help to understand the following statement : The space $L^2=L^2(S_1)$ where $S_1$ denotes the unit circle is the Hilbert space of all the square integrable functions on $S_1$, with ...
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0answers
35 views

Suppose there is a constant $C$ such that $\| f_n - f\|_1 \leq \frac{C}{n^2} $ for all $n \geq 1$. Show that $f_n \rightarrow f$ a.e. [duplicate]

Let $m$ be Lebesgue measure on $\mathbb R$ and let $f_n ,f \in L^1 (m)$. Suppose there is a constant $C$ such that $\| f_n - f\|_1 \leq \frac{C}{n^2} $ for all $n \geq 1$. Show that $f_n \rightarrow ...
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2answers
67 views

Additivity of the union of Lebesgue measurable sets [closed]

Please help me with this!! I dont have any idea how to use the information given in the hypothesis: If $(A_{n})_{n\ge 1}$ is a family of measurable Lebesgue sets, with $\lambda(A_{n}\cap A_{m})=0$, ...
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0answers
39 views

Interior of a Minkowski sum satisfies Brunn-Minkowski inequality.

If $A,B$ are Lebesgue measurable sets in $\mathbb{R}^n$ with $0 <\lambda(A),\lambda(B)< \infty$. Prove that $\lambda (\text{Int}(A+B))^{1/n} \ge \lambda(A)^{1/n}+ \lambda(B)^{1/n}$ where ...
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0answers
13 views

An exercise on image measure

Given: $((0,1), \mathbb B((0,1)), \lambda)$: a measure space, where $\lambda$ is Lebesgue measure. $\mu$: a probability measure on $(\mathbb R, \mathbb B(\mathbb R))$ $F(t) = \mu((-\infty,t])$, ...
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1answer
34 views

Exercise about measurable and continuous functions

I want to propose to you this exercise. Let $f:[0,1]\times \mathbb{R}\to\mathbb{R}$ a function with these properties: 1)For every $x\in\mathbb{R}$ the map $t\mapsto f(t,x)$ is measurable. 2)For ...
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25 views

Is every strongly measurable function to $\mathbb{R}$ also measurable?

Here seems a simple enough proof of the statement: Let $f: X \to \mathbb R_{≥0}$ be measurable. Construct $$f_n(x)=\sum_{k=0}^{n^2}\frac{k}{n}\cdot ...
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3answers
15 views

Outer Measure definition

In the definition of Lebesgue outer measure/ outer measure , m*(A) = inf [Σ l(In)] Here how can one take infimum over a summation? Please elaborate.
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4answers
76 views

If $ f \rightarrow c$ then prove $\frac{1}{a} \int_{[0,a]} f \rightarrow c$

Let $f$ be an extended real-valued $\mathcal{M}_{L}$-measurable function on $[0,\infty)$ such that $f$ is $\mu_L$-integrable on every finite subinterval of $[0,\infty)$, and $$ \lim_{x\rightarrow ...
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1answer
47 views

Properties of mollification

We have this theorem For any $1\le p<\infty$ and $f\in L^p(\mathbb{R}^k)$, then $\|f*\phi_\delta - f\|_p\to 0$ as $\delta\to0$, where $\phi$ is any nonnegative measurable function on ...
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1answer
25 views

I do not understand a point in the proof of completness of $L^{\infty}$

do not understand a point in the proof of completness of $L^{\infty}$. I have this proof. We consider the sets $$A_{n,m}=\{x\in E:|f_{n}(x)-f_{m}(x)\|\leq\|f_{n}-f_{m}\|_\infty\}$$ for all ...
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1answer
41 views

Book Recommendation for Measure Theory in n-Space

What's a standard book on multidimensional measure theory? I'm aware of some books on functions of several variables, but they do not discuss measure theory or Lebesgue integration in space. Thanks. ...
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34 views

Lebesgue integral of a difference of functions.

I would like to prove that if $ \delta $ is close enough to zero then Lebesgue Integral: $ \int_{R} {}|f(x) - f(x + \delta)| dx $ is less then $ \epsilon $ for any $ \epsilon>0 $ ,where f belongs ...
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1answer
27 views

$\mu \ll m$ finite Borel implies $x \mapsto \mu(A + x)$ is continuous

Why is it true that if $\mu$ is a finite Borel measure on $\mathbf{R}$ which is absolutely continuous with respect to Lebesgue measure $m$, then $x \mapsto \mu(A + x)$ is continuous for any fixed ...
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1answer
17 views

Showing a linear properties of outer measure(Lebesgue).

For a subset $A$ of $\Bbb R$ and real numbers $a$ and $b$ define the set $$aA+b=\{ax+b:x\in A\}$$ Show that $m^{*}(aA+b)=|a|m^{*}(A)$ and if $A$ is Lebesgue measurable so is $aA+b$. I don't know how ...
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1answer
34 views

Can one divide a set into subintervals [duplicate]

Sorry that the question title is unclear, I didn't know how to ask it. Take set $A \subseteq [0,1]$, measurable. Does there exist a sequence $x_1,x_2,\dots$ such that $\forall x_i$, \begin{align*} ...
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3answers
43 views
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0answers
22 views

Density of a measure with respect to another measure

Consider $\mathbb{P}, \mu$ measures on the measurable space $(\Omega, \mathcal{F})$. Suppose $\mathbb{P}$ has density $p$ with respect to $\mu$. Let $A \in \mathcal{F}$. Statement: ...
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1answer
27 views

Let $(X, M, \mu )$ be a space with measure. $f:X \to \mathbb R \text{ and } f\in L^1(X).$ ..

Let $(X, M, \mu )$ be a space with measure. $f:X \to \mathbb R \text{ and } f\in L^1(X).$ Prove that for all $\epsilon > 0$ that there exists $\delta > 0$ such that for $E \in M$, $\mu(E)< ...
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0answers
19 views

If $E \subset [0,1]$ satisfies, for any $I \subset [0,1]$, $m(E \cap I) \geq \frac{1}{2}m(I)$, then $m(E)=1$

If $E \subset [0,1]$ satisfies, for any $I \subset [0,1]$, $m(E \cap I) \geq \frac{1}{2}m(I)$, then $m(E)=1.$ I'm aware this post exists elsewhere, say, here but what I don't understand is why we ...
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1answer
37 views

Proving that $f(x)=\frac{1}{x^2 \ln x} $ is Lebesgue measurable on $(2, + \infty)$

I have that a set $E$ is Lebesgue measurable if the outer measure: $$\mu^*(E)=\inf_{I_1,...,I_n} \mu (I), E \subseteq I_1 \cup I_2 ,...\cup I_n , I_i-\text{intervals}$$ satisfy the three properties ...
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1answer
22 views

Show: $\int_{\Omega} f d\mu =\int_{]0,\infty[} \mu(E_t) d\lambda_1(t)$

I have troubles understanding one step in the solution for this task: Let $(\Omega,\mathcal{A},\mu)$ be a $\sigma$-finite measure and $f:\Omega \rightarrow [0,\infty]$ be a measurable function. Let ...
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1answer
47 views

uniform boundedness principle for $L^{1}$

i read this theorem from V.I.Bogachev vol 1 Measure Theory. A family $\mathcal{F}\subset L_{1}(\mu)$,where the measure $\mu$ takes values in $[0,+\infty]$, is norm bounded in $L_{1}(\mu)$ precisely ...
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2answers
50 views

Show that the Lebesgue intergral $\int_{1}^{\infty} x^{-b} e^{\sin {x}} \sin {(2x)}\, dx$ exists iff $b>1$

Assume $b>0$. Show that the Lebesgue intergral $\int_{1}^{\infty} x^{-b} e^{\sin {x}} \sin {(2x)}\, dx$ exists iff $b>1$. We know if $b>1$ the integrand can be bounded and it's just an ...
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1answer
27 views

Sigma algebra examples

So the definition is: Let $X$ be a set. A sigma algebra is a collection $\Sigma \subset 2^X$ s.t. (i) $\emptyset \in \Sigma$ (ii) $E \in \Sigma \implies E^c \in \Sigma$ (iii) $E_1, E_2,... \in ...
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39 views

Will the joint probability density exist in this case?

Assume that we have a probability space $(\Omega, \mathcal{A}, P)$, and we have two random variables $X,Y: \Omega \rightarrow \mathbb{R}$. On this space. We can define two measures ...
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Lebesgue-measurable sets requiring the Axiom of Choice to construct

Every known construction of the Vitali set relies on the Axiom of Choice. It happens to not be Lebesgue-measurable. Must every set whose construction relies on the Axiom of Choice not be ...
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Prove that there exists a sequence of continuous functions $f_n(x)$ such that $f_n \rightarrow f$ pointwise on this interval.

Suppose that the real-valued function $f(x)$ is nondecreasing on the interval $[0,1]$. Prove that there exists a sequence of continuous functions $f_n(x)$ such that $f_n \rightarrow f$ pointwise on ...
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3answers
28 views

Open intervals of zero length are empty

I was reading a solution to a question and it said "As open intervals of zero length are necessarily empty". How can an interval be empty? All I'm imagining is $(0,0)$ which doesn't make sense...
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1answer
54 views

question about the statement on Royden, real analysis 3rd, page 79.

I am very confused about the following statement on Royden, real analysis (3rd), page 79. It follows this proposition that, if $\varphi = \sum^{n}_{i=1} a_i \chi_{E_i}$, then $\int \varphi = \sum ...
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1answer
36 views

Clarification on Measure Theory

My text book says that the Lebesgue measure on Borel $\sigma$-algebra of $\mathbb{R}$ is not complete . I am looking for such a Borel set which has measure $0$ but its subset is not a Borel set. Such ...
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63 views

On generical domains: Riemann integrable $\Rightarrow$ Lebesgue integrable?

If I have correctly understood and been able to generalise the proof that I found in Kolmogorov-Fomin's Элементы теории функций и функционального анализа for the case of $n=1$, I know that, if ...
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1answer
30 views

Example about Fubini/Tonelli Theorems

By using Fubini/Tonelli Theorems, evaluate $$\int^{0}_{1} \int^{1}_{y} x^{-\frac{3}{2}}\cos\left(\frac{\pi y}{2x}\right)dxdy$$ My attempt: by Tonelli Thm $$\int^{0}_{1} \int^{1}_{y} ...
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1answer
48 views

Is it true that the Lebesgue measure of the boundary of any set is zero?

Is it true that the Lebesgue measure of the boundary of any set is zero? If no, then what are the counterexample and what are the conditions under which the above statement is true? Like for example, ...
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22 views

Does $\inf \{|I| : \text{open interval I contains set E} \}=\lambda ^* (E)$

For any bounded $E ⊆R$, let $$L(E) = \inf \{|I| : \text{open interval I contains set E} \}$$ if $E$ is unbounded then let $L(E) = +∞$. Does the outer Lebesgue measure satisfy the property $λ^∗(E) = ...
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0answers
32 views

Prove $λ^∗(U) < ε$

Let $ε > 0$. Prove that there exists an open set $U ⊆ R$ which contains all rational points such that $λ^∗(U) < ε$. Where lambda star is the outer lebesque measure. We know $λ^∗(U) \geq 0$, ...
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1answer
30 views

Finding measure of Cantor set

Let us consider the following function designed to measure the length of subsets of $\mathbb R$: $l^∗ : 2^{\mathbb R} → [0,+∞]$ satisfies $l^∗(∅) = 0$, $l^∗(S) \leq l^∗(T)$, if $S ⊆ T$, ...
3
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1answer
42 views

Borel set that is not countable union or intersection of open or closed sets

In this previous question, one can read the following: It is important to keep in mind, by the way, that Borel sets are more than just countable unions and intersections of open and closed sets. ...