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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If $E$ is Borel, then $\lambda(E)=\sum_{k \in Z} \lambda((E-k) \cap[0,1))$

I'm trying to prove that the measure for a Borel set $E$ can be as $$\lambda(E)=\sum_{k \in Z} \lambda((E-k) \cap[0,1))$$ Where $A-k$ is just a translated set. But I'm having trouble with this. At ...
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Natural density of sets using Lebesgue measure

Suppose that the sets $A$ and $B$ are specified subsets of positive integers up to $n$. (For instance, $A$ or $B$ could be the set of all even integers less than $n$). Assume also that $A$ and $B$ ...
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3 votes
2 answers
80 views

Rudin’s PMA, Theorem 11.20

This is the definition which we need for the theorem: (source) 11.19 $\; \;$ Definition $\; \;$Let $s$ be a real-valued function defined on $X$. If the range of $s$ is finite, we say that $s$ is a ...
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Intersection of meaning between Lebesque Measure and Natural Density

FYI: the articles linked and their content within are well-known in number theory, so number theory is a tag (correct if need be). $\textbf{Background}$: This article by Maier states that a given ...
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10 votes
1 answer
343 views

Can the real numbers be equally split into two sets of same measure?

The rational numbers $\Bbb Q$ are dense in $\Bbb R$, but they are still a set of measure zero, i.e. $$\begin{align} \mu(\Bbb Q \cap [a,b]) &= 0 \\ \mu((\Bbb R\!\setminus\! \Bbb Q) \cap [a,b]) &...
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Whether or not a measure is finite or $\sigma$-finite

I have been given the following question: Let $\mu$ be a Borel measure on $[1, \infty)$ given by the density function 1/x with respect to the 1-dimensional Lebesgue measure. Is measure $\mu$ finite or ...
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1 vote
1 answer
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Evaluate integral wrt Lebesgue measure and find the L^p space if it exists

I have been given the following the question: Consider the following function $f : \mathbb R → \mathbb R$, $$f=2·1_{(-3,1]}-3·1_{[5,+\infty)}$$ Here $1_A$ denotes the indicator function of set A. ...
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2 answers
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Measure theory showing integral of non-negative function is continuous

Let $f:(\mathbb{R}, B(\mathbb{R})\rightarrow(\mathbb{R},B(\mathbb{R}))$ be a non-negative function and $\int_{\mathbb{R}}f d \lambda < \infty$. $F:\mathbb{R} \rightarrow \mathbb{R}, F(x):= \int_{(- ...
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-1 votes
0 answers
58 views

Existence of a function $f:\mathbb{R}^8\to \mathbb{R}$ [closed]

Write $(x_1,x_2,x_3,x_4,x_5,x_6,x_7,x_8)$ for points in $\mathbb{R}^8$. Determine if there exist a function $f:\mathbb{R}^8\to \mathbb{R}$ satisfying all of the following: $f$ is continuous and ...
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3 votes
1 answer
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A critiria of measurability related to Lebesgue Density Theorem

The statement is as follows. Assume $E\subset\mathbb{R}$, $\forall x\in E$, $$\lim_{\delta\to0}\frac{m^*(E^c\cap (x-\delta,x+\delta))}{2\delta}=0$$ where $E^c$ denotes the complement of $E$. Can we ...
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Measurable sets with non computable measure

I am just curious if there are any measurable(in the sense of Lebesque measure) sets with noncomputable measure just as there are noncomputable real numbers. I would be highly obliged for any ...
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1 answer
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Probability measure on the product space that is not the product measure

Suppose you have two probability space $(\Omega, \mathcal{A}, \mu_1)$ and $(\Omega, \mathcal{A}, \mu_2)$. What would be a good measure on $\Omega\times \Omega$, that differs from the product measure (...
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1 answer
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Volume of an interval in $R^N$ seen as the supremum of set of volumes of compact subintervals

In a Lemma to be used in developing properties of Lebesgue Outer Measure on $R^N$ ($N$ is a positive integer) the professor pointed that it is not hard to establish that the volume of an interval $I$ ...
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The measure of an uncountable set [duplicate]

We know that the measure of an uncountable set might be zero for instance the famous Ternary Cantor set. But sometimes it might as obvious. Today I was thinking about some problem and the following ...
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1 vote
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Identity not integrable on surface measure

In my textbook in the definition of center of mass there is a following assumtpion: Let $S \in \mathbb R^m$ be a differentiable manifold such that $l_S(S) < \infty $ and the identity map on $\...
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1 answer
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Is any Borel measurable function in the function space $\mathscr{L}^2$?

I want to prove the relation $$\tag{3} \left|\int_{\mathbb{R}} \tilde{u}\tilde{v}\ d \lambda\right| \leq\left(\int_{\mathbb{R}}|\tilde{u}|^{2}d \lambda\right)^{\frac{1}{2}} \cdot\left(\int_{\mathbb{R}}...
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0 answers
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A space of Lebesgue measure $0$ homeomorph to a space with non-null Lebesgue measure?

Question : Does there exist a space of Lebesgue measure $0$ homeomorph to a space with non-null Lebesgue measure ? My attempt : The problem is I have no idea whether it is true or false. If I were to ...
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0 answers
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Let $f: \mathbb R → \mathbb R_+$ and $g:\mathbb R →\mathbb R_+$ be two Borel measurable functions. Show that: [closed]

I have been given the following question: "Let $f: \mathbb R → \mathbb R_+$ and $g:\mathbb R →\mathbb R_+$ be two Borel measurable functions. Show that: $$(∫_\mathbb R f(x)g(x)/x^2dλ(x))^2 ≤ ∫_\...
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-1 votes
0 answers
32 views

Let $F, G$ be two CDFs and $\epsilon >0$. If $\int_0^tF(t)dt = \int_0^tG(t)dt$ for all $t \in [0, \epsilon]$ is $F(t) \equiv G(t)$ on $[0, \epsilon]$?

Suppose that $F$ and $G$ are two CDFs with continuous distribution functions w.r.t. Lebesgue measure. Then if for some $\epsilon > 0$ the integrals $\int_0^tF(t)dt$ and $\int_0^tG(t)dt$ are equal ...
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2 votes
1 answer
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Existence of integral, $\int_{[1,\infty)} fd\nu$, with respect to measure, $\nu$.

Consider the Borel measure, $\nu$, on $[1,\infty)$ given by $$\tag{1} \nu(A):=\int_A \frac{1}{x}d\lambda, $$ where $\lambda$ is the $1$-dimensional Lebesgue measure. Let $f:[1,\infty)\rightarrow \...
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1 answer
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Is the measure $\nu(A):=\int_A \frac{1}{x}d\lambda$ $\sigma$-finite.

Consider the Borel measure, $\nu$, on $[1,\infty)$ given by $$ \nu(A):=\int_A \frac{1}{x}d\lambda $$ where $\lambda$ is the 1-dimensional Lebesgue measure. I want to know if this measure is $\nu$-...
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0 votes
1 answer
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Help understanding the definition of a specific measure.

In an exercise, I am supposed to determine if a measure is finite or $\sigma$-finite, however, I do not understand the definition of said measure. The measure is defined as a "Borel measure on $[...
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1 vote
2 answers
49 views

open and closed set in $\mathbb{(L_1(\mathbb{R^n})}$

In a test I had was written if $X=(f\in\mathbb{L_1}\mathbb{R^n}|m(f^{-1}((0,\infty))=0)$ is open or closed in $\mathbb{(L_1(\mathbb{R^n})}$. I suspect that this set is none but I am not sure. Is that ...
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2 votes
0 answers
36 views

How to prove that normed space is complete?

$I=[0,1]$. For $k \in \mathbb{N}$, denote by $C^k(I)$ the space of real-valued functions on $I$ possessing continuous derivatives up to order $k$ on $I$, including one-sided derivative at the end ...
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A interesting question about relation between integral of composition(log) of a function and composition(log) of integral of the the function??

Let f:X-->R is a nonnegative measurable function and logf is integrable over X.(where X is a measure space) then to show this following two problem..i am completely stuck in this two problem how to ...
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1 vote
1 answer
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Does $f(x)=\boldsymbol{1}_{(-9,3]}-2\cdot\boldsymbol{1}_{[11,\infty)}$ belong to $\mathcal{L}^{\infty}(\mu)$ with $\mu$ being the Lebesgue measure?

Consider the function $f:\mathbb{R}\rightarrow\mathbb{R}$, $$\tag{1} f(x)=\boldsymbol{1}_{(-9,3]}-2\cdot\boldsymbol{1}_{[11,\infty)} $$ I want to know if the function belongs to the function space $\...
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1 vote
0 answers
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The Lebesgue measure of a zero set of a $C^{3}$ function

Let $f\in C^{3}([0,+\infty))$ be a positive strict decrease funcition satisfying $f(0)=2$ and $$ \lim_{r\to +\infty}e^{r}f(r)=1 $$ For any given $k\geq 4$, Suppose that $\mu_{1},\ldots, \mu_{k}$ are ...
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0 votes
1 answer
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Measure Theory question about Outer measure vs measure

We are currently covering the last chapter of baby Rudin where he quickly covers the basics of Measure Theory. I am having a hard time understanding under what circumstances does the outer measure ...
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0 votes
2 answers
37 views

Confusion regarding Lusin's theorem.

We have measure theory in this semester.I found the statement of Lusin's theorem on the internet to be: Let $f:\mathbb{R\to R}$ be a Lebesgue measurable function.Then for each $\epsilon>0$ there ...
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1 vote
1 answer
47 views

Can we test whether a density is bounded away from $0$ at a point given sample data?

Suppose $X$ is a random variable with unknown support $S \subseteq \mathbb{R}$ and unknown density $f$. Given a finite iid sample $D$, is there a test to check whether the density $f$ is bounded away ...
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4 votes
0 answers
36 views

Derivative of a map $f:\mathbb R \to \ell^2 $ to a separable Hilbert space vs derivative of each component of the Hilbert basis.

${\newcommand{\R}{\mathbb{R}}}$ Let $\ell^2 $ be the Hilbert space of square summable sequences of real numbers. Consider a map $f: \R \to \ell^2$, that has components $f_n:\R\to \R$, i.e. $f(t)=\{...
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0 votes
1 answer
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Does $e^{-\delta t}f(t)\to 0$ if $f \in L^2([0,+\infty))$?

A function $f:[0,+\infty) \to \mathbb R$ in $L^2([0,+\infty))$ does not have to decay at infinity i.e. $f(t) \to 0$ as $t\to +\infty$, even if $f$ is continuous (a sufficient condition proposed in ...
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1 vote
1 answer
41 views

Existence of a collection of balls that cover a zero measurable set with this sum of measures less than $\varepsilon$

Let $\lambda$ the lebesgue measure on $\mathbb{R}^n$. If $N$ is a set with zero measure, then for all $\varepsilon>0$, exists a numerable collections of open balls ${B_n}$ such that $N\subset\...
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1 vote
0 answers
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Properties about the comeagre null sets in $\mathbb{R}^n$ that is non-Borel

I have learnt from this question that comeagre null sets (sets with Lebesgue measure zero whose complement is a countable union of nowhere dense sets) do exist, and they cannot be included in $F_\...
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1 vote
0 answers
73 views

An interesting qn came in my mind related to outer measure of the cartesian product of A and B??

So here are the questions came in my mind, but I could not answer each of them.. If $A$ and $B$ are two subsets of $\Bbb{R}$ then $\lambda_2^*(A×B)=\lambda_1^*(A) \lambda_1^*(B)$?? (If not the case, ...
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2 votes
1 answer
53 views

Derivative(s) of Cantor Measure (Donald L. Cohn ch. 6.2, exercise 6.2.4, related to lemma 6.2.5)

First, context: I'm doing a course on measure/integration theory following the book by Donald L. Cohn. In section 6.2, he defines the (upper and lower) derivates of a finite Borel measure $\mu$ on $\...
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1 vote
1 answer
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Cover of a set of measure zero inside of an open set

Let $U$ be a subset of an open set $A\subset \Bbb{R^n}$. Suppose also that $U$ has measure zero. Then for all $\epsilon>0$, there exists a countable collection of rectangles $\{R_i\}$ that covers $...
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1 vote
1 answer
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Is almost everywhere equality preserved under integration?

Suppose we have four Lebesgue measurable functions: $$ f_1: \mathbb{R} \times \mathbb{R} \rightarrow \mathbb{R} \qquad f_2: \mathbb{R} \times \mathbb{R} \rightarrow \mathbb{R} \\ g_1: \mathbb{R} \...
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1 vote
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Is the solid of revolution obtained by rotating $E\subset OZY$ Lebesgue-measurable when $E$ is Lebesgue-measurable?

Let $E\subset OZY$ a Lebesgue-measurable set in $\mathbb{R}^2$. How can I prove that the solid of revolution obtained by rotating $E$ around the $Z$ axis is a Lebesgue-measurable set in $\mathbb{R}^3$?...
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0 votes
0 answers
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Two elementary questions about integration with respect to measures

I am a self-studying a book about fractal geometry in which some notions of measure theory are reviewed. As one who had zero background in measure theory, I am puzzled about some aspects of what I ...
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1 answer
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Existence of an element $\xi\in \mathbb R$ such that $m(A\cap (B+\xi))>0$

Let $A$ and $B$ are two positive Lebesgue measurable sets in $\mathbb R,$ that is, $m(A)>0$ and $m(B)>0$, where $m$ denotes the Lebesgue measure in $\mathbb R.$ I want to show that, there exists ...
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2 votes
1 answer
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If two functions are always equal a.e. in one argument, are they always equal a.e. in the other argument?

Suppose we have two Lebesgue measurable functions: $$ f: \mathbb{R} \times \mathbb{R} \rightarrow \mathbb{R} \qquad g: \mathbb{R} \times \mathbb{R} \rightarrow \mathbb{R} $$ such that for any $x \in \...
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2 votes
0 answers
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How to show that intersection of crazy sets is crazy?

We call a set $A\subseteq\mathbb R$ to be crazy if for every positive $\delta$ there exists an open set $U$ (with respect to the Euclidean distance on $\mathbb R$) containing $A$ such that the ...
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0 votes
1 answer
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Am I validly applying integration by parts to cumulative distribution functions?

I know there are lots of questions like this question, but I think the question I have is pretty basic, and I could imagine this formulation is useful. Suppose $G$ is a distribution function with ...
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0 votes
0 answers
21 views

Integration and Measure problem from Shilov and Gurevich Book.(zero measure definition using step functions))

The problem reads: Let F be the closed interval obtained by removing a countable collection of disjoint open intervals from a closed interval [a,b], where the sum of the lengths of such intervals is b-...
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0 votes
1 answer
24 views

Prove that $\sup_x |\rho(x)-\rho_{\alpha}(x)| \leq |\rho'|_{L^{\infty}(\mathbb{R})} |1_{x>0}-h_{\alpha}|_{L^1}$

Let $\rho(x)$ be a function such that its derivative is bounded and its limit at $x \to -\infty$ exists. We can rewrite $\rho$ like so and define $\rho_{\alpha}$: $$ \rho(x)=\rho(-\infty)+\int_{-\...
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0 votes
1 answer
43 views

Limit of integral to integral of limit

Given $E \subset \mathbb{R}^n$ measurable and $f:E \rightarrow \mathbb{R}$ measurable and positive then $\lim_{n\rightarrow\infty}{\int_E (f(x))^{\frac{1}{n}} dx} = m(E)$ I feel like id need to apply ...
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0 votes
0 answers
27 views

Composition by exponentiation of measurable functions

A measurable function is defined the following way: $f:X \rightarrow [-\infty, +\infty]$ such that $f^{-1}((-\infty, a))\in \mathcal{M} \ \ \forall a \in \mathbb{R}$ Then, there is this following ...
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0 votes
0 answers
35 views

Image of a set of measure zero by a Lipschitzian function.

Lipsichitzian functions are defined the following way: $f:A\subset R^n \rightarrow R^n$ where $\exists K \geq 0$ such that $||f(x)-f(y)||\leq K||x-y||$ Given this, proove the following: Given a set ...
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1 vote
0 answers
30 views

An inequality of n-dimensional measure

Background the measure of $n$-dimensional simplex A simplex is a generalization of the notion of a triangle or tetrahedron to arbitrary dimensions.More formally, suppose the$ n + 1$ points ${ a_{0},\...
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