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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Counterexamples in Analysis

I want to (dis)prove the following statement: A sequence of functions which converges almost uniformly implies uniform convergence for that sequence of functions. I'm sure I've read up on a ...
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Preimage of open set is Lebesgue measurable

It is a simple result in my book saying the proof is trivial, but I can not seem to show it. If someone can provide a hint just to help me begin my proof, it would be of assistance. Assume you know ...
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$\{A \subset X: \chi_A \in \mathcal{F}\}$ is a sigma algebra

Suppose $\mathcal{F}$ is a collection of real-valued functions on $X$ such that the constant functions are in $\mathcal{F}$ and $f + g$, $fg$, and $cf$ are in $\mathcal{F}$ whenever $f$, $g \in \...
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Lebesgue outer measure is countably subadditive but not finitely additive proof

I have read all the Qs on this but couldn't find a clear proof. How can I prove that Lebesgue's outer measure is not finitely additive? Thanks! Edit: I understand I must show that the measure of the ...
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1answer
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Infinities on null sets

This is a conceptual question! Why is it that (e.g.) $\int_0^1 \frac{1}{x} dx$ doesn't converge. I'm stuck in the following way of thinking about it: Since the problematic part is $\int_0^\epsilon \...
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1answer
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Dilation convergence in L^1

Below is a question, which I asked before, from Stein's Real Analysis. I've provided a partial solution, which I think it's pretty along the lines of what needs to be done, however, I have no ...
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3answers
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Is it correct to interpret the “dx” in the standard notation for integrals as the Lebesgue measure?

Ok so I am in my Calc I class for the summer and we are just beginning to talk about integrals. I know a little bit about measure theory and the Lebesgue integral and why is it more general than the ...
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Diffuse-like decomposition of the segment $[0,1]$ in accordance with Lebesgue measure

Consider the segment $[0,1]\subset\mathbb{R}$ and the standard Lebesgue measure $\mu$ on $\mathbb{R}$. I wonder if we can find such decomposition $A\sqcup B=[0,1]$, that for any subsegment $[a,b]\...
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For any measurable set $A\subset\mathbb{R}$ and $r\in(0,\mu(A))$ we have $(\mu|_{2^A})^{-1}(r)\neq\emptyset$

Recently when I tried to prove a statement I needed to rely on the following fact that intuitively feels correct, but I wasn't able to prove it accurately. Here it is: Consider a set $A\subset\...
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Equivalent definition of Lebesgue measurability in terms of additivity?

When introducing measurability, we noted that we wanted the following property to hold for $A, B \in \mathcal{P}(\mathbb{R})$ $m(A \cup B) = m(A)+m(B)$ (additivity) We then defined a set A to be ...
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1answer
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$m_*(E)=m^*(E)\iff E$ Lebesgue measurable

Let $E\subset [a,b]$. Show that $E$ is Lebesgue measurable if and only if the Lebesgue outer measure of $E$ is equal to the Lebesgue inner measure of $E$. I have seen the proof for this above ...
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dominated theorem

If $\phi(t)$ and $\psi(t)$ are fundamental matrices of differential equations $ dX(t)=A(t)X(t)dt$ and $ dX(t)=B(t)X(t)dt $ If $ g(t)$ is a bounded and measurable function then is it correct to ...
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1answer
81 views

Zero integral implies zero function almost everywhere

Assume $f$ is Riemann integrable and further assume that $\int_a^x f=0$ for all $x$. How would I go about showing that $f$ itself is $0$ almost everywhere? I am new to Lebesgue's measure theory so I ...
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Prove that $0\leq \varphi_n\leq\varphi_{n+1}\leq f$ and that $0\leq f-\varphi_n\leq 2^{-n}M$.

The following is from Carothers' Real Analysis: Suppose $f$ is a nonnegative, bounded, Lebesgue measurable function on $[a,b]$ with $0\leq f\leq M$. Let $E_{n,k}=\left\{\frac{kM}{2^n}\leq f < \...
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Addition of two measurable sets

Notation: $ A+B = \{ a + b : a \in A, b \in B \}. $ H. Steinhaus proved the classical result that $ A+B $ contains an interval if $ A $ and $ B $ are both measurable subsets of the real line, each ...
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Intermediate Value Like Property for Lebesgue Measure

Below is a question from N.L. Carother's book Real Analysis. I've provided my attempt at a solutions, however, any feed back would be very appreciated. Suppose $E$ is a measurable subset of $\...
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Does the image of positive measure set under homeomorphism also have positive measure?

Say I have a homeomorphism $f:A\longrightarrow B$ between open subsets $A$ and $B$ of $\mathbb{R}^n$. If $S\subset A$ has positive Lebesgue measure, does $f(S)$ also have positive measure? If so, do ...
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Alternative definition of Lebesgue measurable set.

Consider $\mathbb{R^d}$ with Lebesgue measure $\mu$. Suppose that for any $\epsilon >0$, there exists a Lebesgue measurable set $F$ such $\mu^* (E\Delta F) < \epsilon.$ Then $E$ is Lebesgue ...
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If function is measurable on an interval, is it measurable on its subinterval?

This is exercise 2.3 from "A User-Friendly Introduction to Lebesgue Measure and Integration" by Gail S. Nelson: Let $[c,d]\subseteq[a,b]$. Show that if $f$ is measurable on $[a,b]$, then $f$ is ...
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1answer
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Sequence of subsets $E_n$ of $[0, 1]$ with $m(E_n) = 1$ for every n but $m(\cap_{n=1}^\infty E_n) \neq 1$

This is actually a "prove or give counterexample" type of problem. The claim is that if any sequence of subsets $E_n$ of $[0, 1]$ have $m(E_n) = 1$ (m being the standard Lebesgue measure) for every n, ...
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Weak convergence of finite measure preserving transformations

I am reading King's paper "The commutant is the weak closure of the powers, for rank-1 transformation" and I am not able to show that: (0.1) "If the $T_i$ are invertible measure preserving ...
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1answer
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$f : B \to \mathbb R^n$ such that $\mathcal L^n(N)=0 \implies \mathcal L^n(f(N))=0$

Let $B$ an open ball in $\mathbb R^n$. Let $f:B \to \mathbb R^n$ measurable and satisfy the property that $N \subset B, \mathcal L^n(N)=0 \implies \mathcal L^n(f(N))=0$, where $\mathcal L^n$ is the ...
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The Lebesgue measure of the set of horizontal lines through the points of a subset $A$ of $\mathbb{R}$ with $\lambda(A)=0$

Suppose $A$ is a subset of the real line with $\lambda(A)=0$ and $H=\{(x,y):x\in A\}$. What is a natural idea behind proving that $\lambda(H)=0$ ? In fact, I wish to prove that the collection of ...
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+100

Minimum of $F$ over Finite Perimeter Sets in $\mathbb R^N$

Problem: Let $G$ be a bounded Borel set. Let $X$ be the set of finite perimeter sets in $\mathbb R^N$ and $F: X \to \mathbb R \cup \{+\infty\}$ defined as \[ F(E)= \begin{cases} Per(E) \hspace{1,...
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1answer
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Change of Variable Proof in Folland

I am reviewing Folland's proof of the following standard result and I have a question on one part. Suppose $\Omega$ is an open set in $\mathbb R^{n}$; $G:\Omega \to \mathbb R^{n}$ is a diffeomorphism ...
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1answer
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How to solve Lebesgue on a set

First I'll have to mention this is not homework or something like that. It's training for an exam and I couldn't find resources on how it is done. If you have any URL showing how to solve these kinds ...
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When we can permute between the integral and convex hull?

Is there a relation between the following expressions? $$\operatorname{conv}\left(\int_{0}^{t} f(s,x)ds :x \in A \right) $$ and $$\int_{0}^{t} \operatorname{conv}(f(s,x):x \in A)\ ds $$ where $A$ ...
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Absolute continuity and continuity

Suppose we have a measure $\mu$ on $(a,b]$ such that $\mu(a,b]=F(b)-F(a)$ where $F$ is non-decreasing, continuous function from the right, Definition: A function $F$ is said to be absolutely ...
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How to show that the measure of $\cap_n E_n $ is not $0$?

Let $E_n$ ($n \in \mathbb{Z}_{\geq 1}$) be the union of a finite set of closed intervals and the sum of the lengths of the intervals is large than or equal to a fixed positive number $a > 0$. ...
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1answer
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Upper and lower bound on $L^1$ norm purely in terms of measure

Suppose $f$ is a measurable almost everywhere finite function on $\mathbb{R}^d$, and let$$E_n = \{x : 2^n \le |f(x)| < 2^{n + 1}\}, \quad n \in \mathbb{Z}.$$What is a non-trivial upper and lower ...
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$L^1$ approximation by a slightly “displaced” copy

Let $f:\Bbb R\to \Bbb R$ be an $L^1$ function and $f_\epsilon(x):=f(x+\epsilon)$, $\mu$ is the Lebesgue measure, prove that $$\lim_{\epsilon\to 0}\int|f_\epsilon-f|\mathrm d\mu=0.$$ I tried to ...
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Two questions on measurable sets.

I'm learning about measure theory, specifically measurable sets, and need help with the following two questions: $(1)$ For $n \in \mathbb{N}$, let $E_n = \{x \in [0, 2\pi] : \sin x < {1 \over n}...
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Does a set with strictly positive Lebesgue measure contain an interval?

I am studying a function whose Fourier transform is zero on a set of strictly positive Lebesgue measure and I need to know this: If a set has a strictly positive Lebesgue measure can we prove that it ...
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Lebesgue outer measure equals Lebesgue Inner Measure

Definition. (Lebesgue Measurable) A set $E$ is said to be Lebesgue measurable if there exists an open set $G$ and a closed set $F$ such that $F\subset E\subset G: m^*(G\setminus F)<\epsilon$. ...
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Existence of a Lebesgue measurable set

The following is from Carother's Real Analysis: Suppose that $E$ is Lebesgue measurable with $m(E)=1$. Show that there is a measurable set $F\subset E: m(F)=1/2$. Carothers offers a hint which ...
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1answer
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Lebesque measurable “sparse” set.

Recently, out of curiosity, I looked up the list of questions for Princeton generals, and one caught my attention: Can you construct a measurable set on the interval $[0; 1]$ such that its ...
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Bounding $L^p$ norm of a function defined by averaging

Let $\Delta=\{t_0, t_1, ... t_m\}$ be a partition of $[a, b]$ and let $f{\in}L^{p}[a, b]$ for $1\le p\le\infty$. Let $T\Delta$ be the function on $[a, b]$ defined by $T\Delta(f)(a)=0$ and $$T\Delta(f)...
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Prove that $\lim_{n\rightarrow \infty} \int_{0}^{1} (f(x))^n dx$ exists if and only if $\mu ( \{x \in [0,1]: f(x)>1 \})=0$

I am trying to solve this problem, but I have not succeeded. Can someone help me? Thanks in advance. The problem is: Let $f$ be nonnegative measurable function on $[0,1]$. Prove that $lim_{n\...
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Lebesgue measure of a sphere

While reading proofs (for ex. this) about measure theory I am inclined to think that it is implicitly intended that the $n$-dimensional Lebesgue measure of a hypersphere $\mathbf{S}^{n-1}$, i.e. of ...
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About measures which respect half-spaces

I was wondering if there are measures known on $\mathbb{R}^n$ which somehow "nicely" see the half-spaces. I am not sure how to exactly quantify "nicely" and hence feel free to make your own ...
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Can distribution function of $f$ be expressed as triangle inequality form?

Let $w_f$ be a distribution function of $f$ on $E\in\Bbb{R^n}$,$$w_f(\alpha) := \mu\left(\{\mathbf{x}\in E ~|~ f(\mathbf{x} \gt \alpha\}\right)$$ From triangle inequality, $$|f| \le |f-f_k|+|f_k|.$$...
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Prove or disprove the existence of a measurable set 'equally' distributed in [0,1].

Is there a measurable set E in $[0,1]$ such that for every open interval $I$ in $[0,1]$, we have $m(E\cap I)=m(I)/2$ where $m$ denotes the Lebesgue measure? Intuitively I think such a set exists and ...
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Intersection of Borel sets with positive Lebesgue-Borel measure [closed]

Let $A \in \mathfrak{B}(\mathbb{R})$ be a Borel set with $\lambda(A) > 0.$ Are there $B, C \in \mathfrak{B}(\mathbb{R})$ with $ B\cap C =\emptyset, B\cup C = A$ and $\lambda(B), \lambda(C) > 0?$...
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Three-dimensional Lebesgue-measure

How can I compute $\int_B gd\lambda^3$ where $$g(x,y,z)=xyz$$ and $$B=\{(x,y,z)\in\mathbb R^3\vert x,y,z ≥ 0, x^2+y^2+z^2 ≤ R^2\},$$ $R>0$ arbitrary? I have no clue on how to find the upper and ...
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Proof the Lemma for use Jordan's Theorem

Show the Lemma following: Let the function $f$ be of bounded variation on the closed, bounded interval $[a,b]$. Then $f$ has the following explicit expression as the difference of two increasing ...
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Why is Caratheodory's characterization of measurability important?

My professor repeatedly emphasizes the importance of Caratheodory's theorem about characterization of measurability, but I don't get why it's so important. As far as I remember, I have never used this ...
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An example of outer measure.

First a few definitions: 1.5.1: Definition. Suppose that $\mu$ is a nonnegative set function on domain $\mathcal{A} \subset 2^X$. A set $A$ is called $\mu$-measurable if for any $\epsilon>0$, ...
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Determining null sets with Tonelli's theorem

How can I show that the diagonal $D=\{(x,y)\in\mathbb R^2\vert x=y\}$ is a Lebesgue-nullset in $\mathbb R^2$ by utilizing the theorem of Tonelli? My solution so far, but it doesn't seem quite right: ...
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Computing a three-dimensional Lebesgue measure of a bounded set

How can I compute the three-dimensional Lebesgue-measure of the set $A$ which is bounded by the areas $x+y+z =6$, $x=0$, $z=0$ and $x+2y=4$? A hint on how I solve problems like this in general would ...
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2answers
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Showing that $f$ is not measurable

I'm learning about measure theory, specifically measurable functions, and need help with the following problem: Let $N$ be a non-measurable subset of $[0, 1]$ and define $$f(x) = \begin{...