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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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Bounded linear maps and Lebesgue integral

Let $L_1([0,1],m)$ be the Banach space of $\mathbb{K}$-valued integrable functions with respect to Lebesgue measure $m$, equipped with the norm $\|f\|_1=\int_{[0,1]}|f|\,dm$, for $f \in L_1([0,1],m)$. ...
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1answer
25 views

Tchebyshev inequality for Measure theory.

Suppose $f$ integrable on measurable set $E$ and $f(x)$ $\geq$ $0$ on $E$, And for any $a \in \mathbb{R}^+$, we define: $E_a$ = $\{x \in E$ $\vert$ $f(x) > a\}$. (which is measurable by basic ...
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0answers
19 views

Question about measure on product space

Consider the set $[0,1]^2$ equipped with a measure $t$ such that the projections of $t$ onto each $[0,1]$ are $\mu$ and $\nu$, respectively, both of which we take to be the Lebesgue measure.. ...
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28 views

Prove that $\phi_{h}(x) = \frac{1}{2h}\int_{x-h}^{x+h}f(t)dt \in L(\mathbb{R})$

Problem. Given $f \in L(\mathbb{R})$, let $$\phi_{h}(x) = \frac{1}{2h}\int_{x-h}^{x+h}f(t)dt,\quad h>0.$$ Prove that $\phi \in L(\mathbb{R})$ e $\displaystyle \int_{\mathbb{R}}|\phi_{h}(x)|dx \...
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1answer
44 views

Show that the open interval (a, b) is Lebesgue measurable

I have to show that an open interval in the form $(a,b)$, where $a,b \in {\mathbb R}$ and $a < b$ is Lebesgue measurable. I think I'm supposed to show, that the subset $(a,b)$ is Lebesgue ...
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25 views

How to show that $\mu\{x:|f(x)|\geq\lambda\}\leq\frac{\|f\|^p_p}{\lambda^p}$ for $f\in L^p(X)$ ($1\leq p<\infty$)?

I am trying to answer this question in measure thoery but I am confused on how to proceed. Let $(X,M,\mu)$ be a measure space with $\mu(X)=1$. Suppose $f\in L^p(X)$ ($1\leq p <\infty$) and $\...
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2answers
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Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be a measurable function, is $f \circ \sin$ measurable?

If $f \circ \sin$ is measurable, then we need to show that $(f \circ \sin)^{-1}(B) = \sin^{-1}(f^{-1}(B))$ is measurable for every Borel set $B$, it is sufficient to show that $\sin^{-1}$ takes a ...
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1answer
36 views

Is the function measurable with respect to Lebesgue measure

Let $h: [a,b]\rightarrow \mathbb{R}$ be a continuous function. For every $y\in \mathbb{R}$, denote $$ S(y)=\begin{cases} 0 & \text{, if } h^{-1}(y) = \emptyset \\ \# h^{-1}(y) &\text{, if } h^{...
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16 views

Sufficient condition for $L^\infty$ distance bound

Suppose for two functions $f, g\in L^{\infty}([0,1])$ and $$||f-g||_{L_1([0,1])}\leq \epsilon$$ Under what condition can we get $||f-g||_{L^\infty([0,1])}\leq K(\epsilon)$ where $K(\cdot)$ is some ...
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65 views

If $\int f=0$ then $f=0$ a.e. with $f\geq 0$. Is it true if $f(x) = \infty$?

I have a doubt! I know that if $f$ measurable and nonnegative, $\int f=0$ implies $f=0$ a.e. And if $m(E)=0$ then $\int_{E}f=0$ (even if $f(x)=\infty$ forall $x$) If $f(x)=\infty$ forall $x$, $f:X\...
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0answers
31 views

Proof of a convergence of sets in the context of Finite Perimeter sets

Let $E \subset \mathbb{R}^n$ be a set of finite perimeter that satisfies $ \mathcal{L}^n (E) < \infty$. Assume that $E$ is symmetric with respect to the hyperplane $\{x_n = 0\}$. We know that there ...
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1answer
65 views

Calculating the norm of a specific bounded linear functional

Let $L_1([0,1], m)$ be the Banach space of $\mathbb{K}$-valued (i.e. $\mathbb{C}$ or $\mathbb{R}$-valued) Lebesgue-integrable functions, where $m$ is the Lebesgue measure, be equipped with the norm $‖...
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23 views

Properties of Normal Numbers

I found these two properties of normal numbers on the Wikipedia page for "normal numbers", but I can't find any other source for them. I was wondering if they were in fact true, and if somebody could ...
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1answer
22 views

Showing a sequence of functions $f_n$ does not converge uniformly to $f$ on an interval.

Suppose for each $n \in \mathbb{N}$ we have a function $f_n:[0,1] \to [0,1]$ by $f_n(x)=nx$ on the interval $x \in [0,\frac{1}{n}]$ and $1$ if $x \in (\frac{1}{n},1]$, and define $f=\lim_{n \to \infty}...
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1answer
47 views

Example of a set which is not in the product $\sigma$-algebra

Let $L_d$ be the $\sigma$-algebra of Lebesgue measurable subsets of $\mathbb{R}^d$. By using Vitali's set $E \subseteq [0,1]$, I am looking for an example of $A \in L_2$ which is not in the product $\...
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1answer
23 views

Unicity of the projection on a closed convex subset of $L^p$

The original problem (not full) statement is the following : Let $(\Omega, \mathfrak{B},\mu)$ a measured space and $p>1$ and let $C$ be a closed convex subset of $L^p$. For $u\in L^p$ let $...
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34 views

Is it possible to write the preimage of a Lebesgue integrable function as the union of nonmeasurable sets?

Is it possible to write the preimage of a Lebesgue integrable function as the union of nonmeasurable sets? I guess this is sort of posed below in the opposite way: Let's say $f$ is a lebesgue ...
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a.e. point wise convergence on finite measured set implies convergence in measure

Let $f,f_k$ for $k \in \mathbb{N}$ be measurable finite a.e. on measurable set $E$. Suppose $f_k \rightarrow f$ point wise a.e., then $f_k$ converges to $f$ in measure on $E$. This is a basic result ...
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1answer
30 views

Convergence and Measure

Prove that if $(f_n)$ is a sequence of nonnegative, measurable functions on $[a,b]$ such that $lim_{n\to\infty}\int_a^b f_n(x)dx=0$, then $(f_n)$ converges to $0$ in measure. Show by example that we ...
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1answer
28 views

$\lambda$-almost-everywhere convergence implied by lim$\lambda (${$x \in E : | f_n(x) - f(x) | > \epsilon$}$) = 0$

Let $E \subset \mathbb{R}$ be $\lambda$-measurable and let $f_n,f: E -> \mathbb{R}$ $\lambda$-integrable, so that for all $\epsilon > 0$ $\lambda(${$x\in E: |f_n(x)-f(x)| > \epsilon$}$)->...
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1answer
27 views

Showing $B$ Borel set, then $f^{-1}(B)$ is a Borel set with $f$ continuous function.

Let $f:\mathbb{R}\to [-\infty,\infty]$ continuous function. (a) Let $\Omega=\left\{E:f^{-1}(E) \text{ is a Borel set } \right\}$. Show that $\Omega$ is $\sigma$-algebra. I already proved this. :) (...
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2answers
46 views

Dilatation of a measurable set is a measurable set.

Let $E$ Lebesgue measurable set. Let $\lambda\in\mathbb{R},\lambda\neq 0$. Let $\lambda E=\left\{\lambda a:a\in E\right\}$. Show that $\lambda E$ is measurable. I have this. Let $f:E\to \lambda E$...
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1answer
35 views

Hint on measure theory problem about rational-invariant measurable sets

I'm trying to solve the following problem: Suppose that $A \subset \mathbf{R}$ is Lebesgue measurable and is such that for each $x \in A$, $x + \mathbf{Q} \subset A$. Show that $\lambda(A)$ or $\...
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0answers
17 views

If $\mathcal N$ non measurable, $\{0\}\times \mathcal N$ is measurable in $\mathbb R^2$, but $m(\{0\}\times \mathcal N)$

We define product measure on complete measurable spaces because if $N$ is a nul set and $M$ is measurable, we want that $$m_2(N\times M)=m_1(N)m_1(M)=0.$$ Let $(\mathbb R^2, \mathcal M\times \mathcal ...
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1answer
21 views

Prove there exists a measurable subset $E' \subset E$ such that $f$ is almost everywhere of one sign on $E'$.

Let $f \in L\left(\mathbb{R}\right)$ where $L$ denotes Lebesgue integrable space of functions. Suppose there exists a subset $E \subset \mathbb{R}$ such that $m(E) > 0$ and $f(x) \neq 0$ for all $x ...
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1answer
12 views

Measurable sets in pratice with Lebesgue-Stieltjes measure

If we consider $F \colon \mathbb{R} \to \mathbb{R},$ defined as $$ F(x) = \left\{\begin{array}{cc} 0, & \mbox{if } x < 0 \\ 3, & \mbox{if } 0 \le x < 4 \\ 8, &\mbox{otherwise.} \...
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1answer
77 views

Prove $L^1(\mathbb{R})\cap L^2(\mathbb{R})$ has empty interior both in $L^1(\mathbb{R})$ and in $L^2(\mathbb{R})$

I suppose it has interior, so we can find an open ball of radius epsilon in $L^1(\mathbb{R})\cap L^2(\mathbb{R})$ and so that ball is contained in both the sets, so we can write: $$B_\varepsilon=\{u(x)...
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2answers
21 views

Continous function with measure of image of zero set positive

I would like to ask you how to find a continuous function f so that for a Lebesgue-zero-set N we get λ(f(N)) > 0 wit λ being the Lebesgue measure. Any chance I can work with the Cantor function? But ...
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1answer
44 views

Showing that a set is Lebesgue Measurable in Higher Dimensions and Applying Fubini's Theorem

I have an idea of how to proceed, but I'm suspicious that my efforts were of no use. Let $A\in\mathcal{M}$ be Lebesgue measurable, and let $g,h:A \rightarrow \bar{\mathbb{R}}$ be Lebesgue measurable ...
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1answer
30 views

Completion of $\mathbb{E}$ by $\mathbb{N}_\mu$ regarding Lebesgue measure - are my arguments valid

I have the measure space $(X,\mathbb{E},\mu)$ where $\mathbb{E}$ is a $\sigma$-algebra on $X$ and $\mu$ is the Lebesgue measure. I have that $$\mathbb{N}_\mu=\{E\subseteq X\; |\; \text{there exist }...
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1answer
34 views

Computing the Lebesgue integral over a ball

Let $\alpha \in \mathbb{R}$ and $\lambda_n$ the Lebesgue measure on $\mathbb{R^n}$. Define $f:\mathbb{R^n}\backslash(0)\to\mathbb{R}, \ f(x)=\left\lVert x\right\rVert^\alpha$ For which $\alpha$ is $...
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1answer
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Why Vitali set as a member of the power set of [0,1] but non-measurable? [closed]

If the sets in a sigma algebra can be called measurable, then the power set of [0,1], as a sigma algebra, in which those sets should be measurable. But Vitali set, as a subset of [0,1], should then be ...
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2answers
20 views

Sequence of Lebesgue integrable functions on R that converges pointwise but for which term by term integration is not valid

Can anyone give an example of a sequence of Lebesgue integrable functions on R that converges pointwise but for which term by term integration is not valid? I know that the Lebesgue Monotone ...
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1answer
18 views

Proving an estimate of $ \mathcal{L}^n (A_1 \cap A_2)$ from estimates of $\mathcal{L}^n (A_1)$ and $ \mathcal{L}^n (A_2)$.

Consider two measurable sets $A_1, A_2 \subset \mathbb{R}^n$ and let $B$ be an open ball in $\mathbb{R}^n$. Assume that $A_1 \subset B$, $ A_2 \subset B$ and that for a certain $0 < \epsilon < 1 ...
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0answers
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Building a Cantor set with positive Lebesgue Measure [duplicate]

Changing the lengths of the intervals excluded during the construction of the ternary Cantor set, show that is possible to build a compact, totally disconnected and perfect set (a Cantor set) with ...
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1answer
39 views

Hardy-Littlewood maximal function $f^*$ is greater than $f$ for measurable $f$

Suppose $f$ is a Lebesgue measurable function on $\mathbb R$, and $\forall x\in \mathbb R$, define the Hardy-Littlewood maximal function $f^*(x)=\sup_{t>0}$$1\over{2t}$$\int_{x-t}^{x+t} |f|$ My ...
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1answer
31 views

Product of a null set

Let $E \subset \mathbb{R}^m$, $F \subset \mathbb{R}^n$ and $\mu$ the Lebesgue measure. How to prove: If $E$ is a Lebesgue null set of $\mathbb{R}^m \Rightarrow E \times F$ is a Lebesgue null set of $...
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1answer
36 views

Is $|x|^{-\alpha}$ integrable for polynomially bounded measures on $\mathbb{R}^n$

We know that $|x|^{-\alpha}$ is in $L^1 (x\in \mathbb{R}^n:|x| \ge 1)$ with the normal Lebesgue measure for $\alpha > n$. But what if we had a measure $\mu$ on $\mathbb{R}^n$ which is polynomially ...
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1answer
235 views

Limit almost everywhere of averages of uniformly bounded and integrable functions .

Let $f_n :[0,1] \to \Bbb{R}$ a sequence of uniformly bounded measurable functions with the property: $$\int_0^1f_n(x)f_m(x)dx=0,\forall m \neq n$$ Prove that $\frac{1}{N}\sum_{n=1}^Nf_n(x) \to 0$ ...
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3answers
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Lebesgue Fundamental calculus theorem

How can I formally show that if $f:[0,1]\to \mathbb{R}$ continuous, $f'(x)$ exists for everything $x\in[0,1]$ y $\sup_x |f'(x)|=B<\infty$ then $\sup_{n,x} |g_n(x)|\leq B$ with $g_n(x)=\frac{f(x+1/n)...
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0answers
27 views

Example for a differentiable function that is not absolutely continuous.

$\mathbf{Theorem:}$ Let $f$ be defined on $I=[a,b]$ and absolutely continuous on $I$. Then $f$ is differentiable a.e in $(a,b)$, and its derivative is integrable over $[a,b]$ with $$\int_{[a,b]}f'=f(b)...
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0answers
24 views

Calculate Lebesgue and Hausdorf measure of a hexagon

Given this hexagon $P$, I've got to calculate the Lebesgue measure $\lambda_{2}(P)$ and the Hausdorff measure $\mathscr{H}^1(\partial P)$. My thoughts are: You can leave out the 6 line segments of ...
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0answers
37 views

Question on strictly positive measure of a level set

Let $f \in W^{2,p}(U)$ for $1 \le p \lt \infty$ where $U$ is a compact subset of $\mathbb R^2$ and $g \in L^{\infty}(U)$ with $g(f)=f$ almost everywhere on $U$. I have trouble understanding why the ...
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1answer
33 views

Finding a Hahn Decomposition involving a Dirac Measure

Let $(X,F,\mu)$ be a finite measure space, i.e. $\mu(X)<\infty$. And let $x\in X$, and let $\delta_x$ be the Dirac measure with respect to $x$, i.e. $\delta_x(E)=1$ if $x\in E$ and $\delta_x(E)=0$ ...
2
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1answer
33 views

Is the product operator lebesgue measurable?

Define $T(x,y):=xy$ for $(x,y)\in \mathbb{R}^2$. Since $T$ is continuous(moreover is of $C^\infty$), it is Borel measurable. However, is $T:\mathbb{R}^2\rightarrow \mathbb{R}$ Lebesgue measurable? ...
0
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1answer
23 views

Every section null implies null in product measure

Let $A\subset[0,1]^2$ be a set such that every section $A_x=\{y:(x,y)\in A\}$ is a null set in $[0,1]$. Can we conclude that $A$ is a null set in $[0,1]^2$? Some context: It is a standard fact that ...
5
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1answer
32 views

Ergodic action of dense subgroup

Let $G$ be a group acting ercodically on a probability measure space $(X, \mu)$. Let $\Gamma$ be a countable dense subgroup of $G$. Is the action of $\Gamma$ also ergodic? The case I am interested in ...
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1answer
44 views

$f$ is continuous on $\mathbb R -\{0\}$, but cannot be made into a continuous function by change on a set of measure zero. [closed]

Consider $f\colon \mathbb R \to \mathbb R$ given by $$ f(x) = \begin{cases} 1 & x \ge 0\\ 0 & x < 0\end{cases} $$ $f$ is continuous on $\mathbb R -\{0\}$, but cannot be made into a ...
1
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1answer
83 views

Equivalence between Lusin (N) property and absolute continuity.

Let $F\colon [a,b] \to \Bbb R$ be continuous and increasing. Then: $F$ satisfies the Lusin (N) property if and only if it is absolutely continuous. Here, Lusin (N) property means that for all $N\...