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Questions tagged [lebesgue-measure]

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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13 views

Measure in the square with uniform marginals are characterized by their support?

Is it true that two measures on the unit square with uniform marginals and the same support are the same measure? If not, can you show an easy counterexample?
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1answer
34 views

Prove $h(y) = \int_{\mathbb{R}} f(x,y) dx$ is continuous if $f(x,y)$ is continuous for every fixed $x$ and $f(x,y)$ is integrable for every fixed $y$.

Define a function $f : \mathbb{R} \times \mathbb{R} \rightarrow \mathbb{R}$. Suppose that for every fixed $x \in \mathbb{R}, f(x,y)$ is continuous. And suppose that for every fixed $y \in \mathbb{R}, ...
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0answers
62 views

Comparing $\sigma$ algebras with topologies

I am learning measure theory from Papa Rudin. I am just trying to capture the ideas conceptually. First of all, how far is measurable sets from topological sets. I guess this question could be ...
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0answers
29 views

Subset of Lebesgue measurable subset of Vitali set is NOT Lebesgue measurable

$B_x = \{y \in [0,1]: x-y \in \Bbb{Q}\}$, $ \varepsilon=\{C \subset [0,1]: \exists x \quad C=B_x\} $. By the axiom of choice we can choose exactly one element of each equivalence class $\varepsilon$ ...
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0answers
25 views

Proof of Lusin's Theorem with the characteristic function

Let $m$ be the Lebesgue measure and the set $E$ be Lebesgue-measurable, and $m(E)<\infty$. Prove that for any $\epsilon>0$ there is a compactly supported continuous function $g:\mathbb{R} \to \...
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0answers
59 views

Direct proof that the irrationals on $[0,1]$ have measure $1$

I am seeking a “direct” proof that the Lebesgue measure of the irrationals on the unit interval is $1$. The standard proof I see is that the measure of the unit interval is 1, and the rationals have $...
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0answers
15 views

Lebesgue measure restricted to subsets of the real numbers.

We have the Lebesgue measure $$\overline{\lambda}: \mathcal{B}(\mathbb{R})_\lambda\to [0, +\infty]$$ where $\mathcal{B}(\mathbb{R})_\lambda$ is the completed $\sigma$-algebra of the Borel sets on ...
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1answer
31 views

Show that $lim \int f_n = \int f$ in a finite measure space

$(f_n)$ is a sequence of integrable functions that converges uniformly to $f$. The question asks to show that if the space has a finite measure, then $lim \int f_n = \int f$. I've tried using the ...
2
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1answer
35 views

$f$ is the pointwise limit of measurable functions $f_n$. Show $\lambda(f^{-1}(a,\infty]) \leq \liminf_{n\to \infty }\lambda(f_n^{-1}[a, \infty])$

Let $A \subseteq \mathbb{R}$ be measurable. Let $f_n : A \rightarrow [-\infty, \infty]$ be measurable $\forall n \in \mathbb{N}$ with $\lim_{n \rightarrow \infty} f_n(x) = f(x), \forall x \in A$. I ...
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1answer
25 views

For all $E$ and $\epsilon>0$ there is $m^*(O)<m^*(e)+\epsilon$

Prove: For all $E\subseteq \mathbb{R} $ and for all $\epsilon>0$ there is an open set $O$, with $E\subseteq O$ such that: $$m^*(O)<m^*(e)+\epsilon$$ I am acquaint with the following: For all $...
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0answers
30 views

Prove $A\subseteq B \subseteq \mathbb{R}\implies m^*(B)-m^*(A)\leq m^*(B \setminus A)$

Prove $$A\subseteq B \subseteq \mathbb{R}\implies m^*(B)-m^*(A)\leq m^*(B \setminus A)$$ where $m^*(B)$ is finite Let $s_i\subseteq t_i$ Be the covers of $A\subseteq B\ $ respectively $1$.$m^*(B)-...
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1answer
41 views

If $E$ has finite Lebesgue measure, then there is an $n\in \mathbb{N}$ such that $\lambda(E) = \lambda(E \cap [-n,n])$. [on hold]

If $E$ has finite Lebesgue measure, then there is an $n\in \mathbb{N}$ such that $\lambda(E) = \lambda(E \cap [-n,n])$. I think the statement is true, but I do not know how to prove it. Thanks!
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0answers
25 views

Suppose that $E \subset \mathbb{R}$. If $E \cap F$ is Lebesgue measurable for all measurable $F$ such that $m(F)$ is finite, then $E$ is measurable.

Suppose that $E \subset \mathbb{R}$. If $E \cap F$ is Lebesgue measurable for all measurable $F$ such that $m(F)$ is finite, then $E$ is measurable. I have spent hours trying to prove this statement. ...
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0answers
65 views
+50

Proof that makes use of the differentiability of a function and of its convex conjugate

I would like your help to understand what are the crucial assumptions driving the claim reported below. Let me start with the notation $\mathcal{Y}\equiv \{1,2,...,M\}$. $S$ is a random vector with ...
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0answers
16 views

Measurable function - characteristic function of measuable set.

Let $g: \mathbb{R^2}\to [0,+\infty]$ Borel measurable function on $\mathbb{R^2}$.If $A \subseteq \mathbb{R^2}$ Borel measurable set prove that function $f$ is measurable:$$f(x)= \int_{\mathbb{R^2}}\...
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1answer
22 views

Lebesgue sigma algebra

A set $E\subseteq \mathbb{R}$ is called measurable if for all $A\subseteq \mathbb{R}$ $$m^*(A)=m^*(A\cap E)+m^*(A\cap E^c)$$ The set of all measurable sets is called Lebesgue sigma algebra ...
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1answer
23 views

Lebesgue measure, there exists a measurable subset $E_t\subset E$ with $m(E_t) = t$.

Let $E\subset\mathbb{R}$ be a measurable set of Lebesgue measure $m(E) = 1$. Then why to any $t\in [0, 1]$, one can find a measurable subset $E_t\subset E$ with $m(E_t) = t$?
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0answers
17 views

Cantor-Like Sets

I'm working on a problem from Stein and Shakarchi's Real Analysis, and it asks to construct a measurable subset $E\subset [0,1]$ such that for any non-empty sub-interval $I$ in $[0,1]$, both $E\cap I$ ...
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1answer
31 views

Measurable functions.

Let $f(x,y)$ be nonnegative Borel function on $\mathbb{R^2}$.Prove that $\psi(x) = \int_{\mathbb{R}}f(x,y)d\mu(y) $ is also Borel function($\mu $ is symbol for Lebesgue measure). My attempt: () $f(x,y)...
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1answer
22 views

For every Lebesgue measure set $E,$ the map $x\mapsto\overline{\lambda}(E\cap(E+x))$ is continuous

I'm trying to prove the next: For every Lebesgue measure set $E,$ the map $x\mapsto\overline{\lambda}(E\cap(E+x))$ is continuous. Here $(\mathbb{R},\mathcal{A}_{\mathbb{R}}^{*},\overline{\lambda})$ ...
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0answers
20 views

If $f:[a,b]\to\mathbb{R}^2$ ($n>1$) is a continuous rectifiable path, then m(f([a,b]))=0 [duplicate]

If $f:[a,b]\to\mathbb{R}^2$ is a continuous rectifiable path, prove that the measure of $f([a,b])$ is null. I've tried to use the fact that $f$ is uniform continuous, but I only got that $m(f([a,b]...
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1answer
59 views

Exercise 36 Ch 1 in Stein's Real Analysis [duplicate]

Construct a measurable set $E\subset [0,1]$ such that for any non-empty open sub-interval $I$ in $[0,1]$, both sets $E\cap I$ and $E^c\cap I$ have positive measure. [Stein's Hint: For the first ...
3
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0answers
48 views

Proof that continuous curve in $\mathbb{R}^2$ has Lebesgue measure zero

Suppose $\Gamma$ is a curve $y = f(x)$ in $\mathbb{R}^2$, where $f$ is continuous. Show that $m(\Gamma)=0.$ [Hint: Cover $\Gamma$ by rectangles, using the uniform continuity of $f$.] My attempt: ...
0
votes
1answer
26 views

Does there exist an open subset $A \subset [0,1]$ such that $m_*(A)\neq m_*(\bar{A})$?

Does there exist an open subset $A \subset [0,1]$ such that $m_*(A)\neq m_*(\bar{A})$? I was thinking we could approximate any set from inside by a closed set . This need not true from outside. So ...
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0answers
45 views

No continuous function $f$ on $\mathbb{R}$ such that $f(x)=1_{[0,1]}(x)$ almost everywhere

Let $1_{[0,1]}$ be the characteristic function of $[0,1]$. Show that there is no everywhere continuous function $f$ on $\mathbb{R}$ such that $$f(x)=1_{[0,1]}(x)\,\,\,\,\,\,\,\,\,\,\,\,\,\text{...
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1answer
16 views

How many events can a space $\Omega$ contain in order for the elementary, discrete definition of expectation to still be valied

Let $(\Omega,\mathcal{A},\mathbb{P})$ be a probability space and $X:\Omega\rightarrow \mathbb{R}$ a random variable on it. Now the definition of expected value would be $$\mathbb{E}[X]=\int _{\...
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1answer
12 views

If $\Omega$ has finite measure and $f \in L^p(\Omega)$ then for $1 \leq q \leq p$ we have $f \in L^q(\Omega)$ and the following estimate.

I was given Hölder's inequality in this form If $1 \leq p \leq \infty$, $1/p + 1/p' = 1$, $f \in L^p(\Omega)$ and $g \in L^{p'}(\Omega)$ then $fg \in L^1(\Omega)$ and $$||fg||_1 \leq ||f||_p||g||...
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1answer
49 views

Is a set of measure zero in $\mathbb{R}$ totally disconnected?

Let $M \subset \mathbb{R}$ be a nonempty set of Lebesgue measure zero. Does it follow that $M$ is totally disconnected in the sense that for any $x<y$, with $x,y\in M,$ there exists $z\notin M$ ...
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2answers
47 views

Existence of a Sequence of strictly positive powers of (1/2) whose sum equals 3/8.

I am writing a proof for my measure theory course that seems to boil down to the existence of a sequence of the form $$ \left\{\frac{1}{2^{n_1}}, \frac{1}{2^{n_2}}, \ldots \right\} $$ with $n_k \in \...
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1answer
56 views

Maximizing the value of an integral

Let $f \colon \mathbb R^N \to \mathbb R$ be a measurable, bounded function. Let $$ \mathcal A := \left\{ g \colon \mathbb R \to [0,+\infty): g \text{ is measurable and} \int_\mathbb R g =1\right\}. $$...
3
votes
1answer
62 views

Prove that integral of infinite sum is in $L^1$

The function $g$ is a composition of functions as follows ($\chi_A$ is the characteristic function of the set $A$): $$f(x) = \frac{1}{\sqrt{x}} \chi_{[0,1]}(x)$$ For an enumeration of the rationals $...
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0answers
17 views

Elementary proof of Steinhaus theorem

I am trying to prove Steinhaus theorem as follows: Let $A,B\in\mathcal{A}_{\mathbb{R}}^{*}$ given with $\overline{\lambda}(A)<+\infty$ and $\overline{\lambda}(B)<+\infty.$ Defining $\overline{\...
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1answer
34 views

Proving that the integral of a function with respect to a measure is finite when the measure is bounded

For some measure space $(X, M, \mu)$ such that $\mu(X)<\infty$. For any measurable function $f: X \to [0,\infty]$ with the property that $$\exists C>0, \exists \alpha < -1, \forall \epsilon&...
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3answers
60 views

$ \mu(|f|\geq \alpha) = \frac{1}{\alpha} ||f||_1$

I'm having difficulty with this problem here: Let $f\in L^1(X,\mathcal{M},\mu)$ with $||f||_1 \neq 0$. Prove that there exists a unique $\alpha$ so that $ \mu(\lbrace|f|\geq \alpha\rbrace) = \frac{1}...
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1answer
22 views

Lebesgue outer measure of image of set is less than or equal to the Lebesgue outer measure of set

For a differentiable function $f : \mathbb{R} \to \mathbb{R}$ with $|f'(x)|\leq 1$ and any set $E \subset \mathbb{R}$ $$m^*(f(E)) \leq m^*(E)$$ where $m^*$ is the outer Lebesgue measure First, the ...
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1answer
124 views

Show m*(𝐴 ∪ 𝐵) = 𝑚*(𝐴) + 𝑚*(𝐵) iff there exists measurable sets A1 and B1 such that 𝐴 ⊂ A1 ,𝐵 ⊂ B1 and 𝑚(A1 ∩ B1 ) = 0.

Q:Let 𝐴 and 𝐵 be given sets of finite outer measure. Show that m*(𝐴 ∪ 𝐵) = m*(A) + m*(B) if and only if there are measurable sets A1 and B1 such that 𝐴 ⊂ A1 ,𝐵 ⊂ B1 and 𝑚(A1 ∩ B1) = 0.Here ...
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1answer
23 views

How to prove that Riemann integrable function is measurable?

If a function $f: A \rightarrow \mathbb{R}$ on a bounded set $A$ is Riemann integrable, how do you prove that $f$ is measurable? From partitions by $n$-dimensional intervals, I can make simple ...
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1answer
20 views

Metric space associated with measure space is path-connected

Im trying to prove the next: Let $(X,\mathcal{S},\mu)$ a measure space and let $\mathcal{F}=\{A\in\mathcal{S}:\mu(A)<\infty\}.$ Defining a relation ~ in $\mathcal{F}\times\mathcal{F}$ as: $A$~$B$ ...
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0answers
21 views

Probability Measure of a Level Set [closed]

Supose we have a $\theta\in \Theta\subset \mathbb{R}^n$ distributed $f(\theta)$ and a function $h:\mathbb{R}^n \rightarrow \mathbb{R}$. I there a expression for the measure of the following sets: $$\...
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1answer
22 views

Basic approximation lemma for Lebesgue sets

I'm trying to prove the next approximation lemma for Lebesgue sets: If $E\subset\mathbb{R}$ such that for all $\epsilon>0$ there is $A=A_{\epsilon}\in\mathcal{A}$ with $\overline{\lambda}(E\...
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votes
1answer
81 views

Limit $\lim_{n\to \infty} n^2\int _0^1\frac{1}{(1+x^2)^n}$

$\lim_{n\to \infty} n^2\int _0^1\frac{1}{(1+x^2)^n}=?.$ My attempt: I want to use Lebesgue Dominated convergence theorem to solve this, Because I see that for $\lim_{n\to \infty}\frac{n^2}{(1+x^2)^n}$...
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0answers
53 views

Lebesgue Outer Measure Limit

Let $\mu^{*}$ be the Lebesgue outer measure on $\mathbb{R}$. I found the following exercise in a textbook. Exercise. For every $A \subseteq \mathbb{R}$, $\lim_{k \to \infty} \mu^{*}(A \cap [-k,k]) = ...
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votes
0answers
27 views

Integral of cantor function

Let $\phi$ denote the cantor ternary function and $\mu = \mu_{\phi}$ the associated measure. Let $\lambda$ denote the lebesgue measure. Compute the following integrals. $\int_{[0,1]} \phi(t)d\mu (t)$ ...
0
votes
1answer
25 views

Lebesgue-measurable or Borel-measurable

In practice, does one always use Lebesgue-measurability? So do we always use Lebesgue-measurable instead of Borel-measurable on $\mathbb{R}^n$ because it is usually more convenient?
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0answers
22 views

Reference request: Bounded function can be approximated by continuous functions in $L_1$ with bounded $L_\infty$-norm

I think that it is well-known that a real valued function $f\in L_\infty[a,b]$ can be approximated by continuous functions $f_n$ with respect to the $L_1$-norm, i.e. $||f_n-f||_{L^1}\to0$, where the $...
0
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0answers
27 views

Show that $\lambda_{F | (F(- \infty), F(\infty)]} = \lambda_{|(F(- \infty), F(\infty)]}$

Let $F: \mathbb{R} \to \mathbb{R}$ be non-decreasing, continuous, and bounded. Let $\lambda_F = \mu_F \circ F^{-1}$, where $\mu_F$ is the finite Borel measure. Show that $\lambda_{F | (F(- \infty), ...
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votes
0answers
28 views

Question on proving $\mathbb{R}$ has non-measurable sets

Show that for every $X \in \mathcal{L}$ with $\lambda(X) > 0$, we have $S \cap X \notin \mathcal{L}$, where $\mathcal{L}$ is the lebesgue $\sigma$-algebra (Lebesgue measurable sets) $S = F + T$ $...
0
votes
0answers
25 views

Explanation of a proof of Steinhaus Theorem - Help to understand what the teacher did

I'm studying Measure Theory and I need help to understand some steps of a proof of Steinhaus Theorem that the teacher did in class. Definition: A cell in $\mathbb{R}^n$ is a set of the form $I=I_1\...
-1
votes
0answers
25 views
2
votes
1answer
40 views

Find a Borel set $F$ such that $\lim_{k \to 0} \frac{m_N(F \cap B_k(x))}{m_N(B_k(x))}$ does not exist at some $x$. [closed]

Let $F$ a Borel set of $\mathbb{R}^N$ and we define, if exists, $$ D_F(x) := \lim_{k \to 0} \frac{m_N(F \cap B_k(x))}{m_N(B_k(x))}, $$ where $B_k(x)$ is the open ball of radius $k$ and ...