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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23 views

Shortest Proof of Lebesgue Dominated Convergence Theroem ( 5 lines) without using Fatou's lemma

If each $G_n\le M $ is a bounded measurable function and $\lim\limits_{n\mapsto \infty} G_n =F$ on a bounded measurable set E , $\epsilon> 0 $ Let $A_n =$ { $x : |F_m(x)-F(x)|<\epsilon$} ...
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0answers
8 views

Outer measure of a nested sequence of non-measurable sets

Let $\bigcup_{n=1}^\infty E_n=E$ and $ E_{n} \subseteq E_{n+1} $ then $\lim\limits_{n\mapsto \infty} \mu^*(E_n) = \mu^*(E) $ even if each $E_n$ is a non-measurable set, where $\mu^*$ is outer ...
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2answers
39 views

Prove that $F(x,y)=f(x-y)$ is Borel measurable

Suppose $A$ is a subset of $\Bbb R$, let $s(A)=\{ (x,y)\in \Bbb R \times \Bbb R :x-y\in A\}$. I already showed: If $A\in \Bbb B$ (Borel measurable set), then $s(A)\in \Bbb B \times \Bbb B$. I want ...
2
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1answer
50 views

If $f:\mathbb{R}\to[0, \infty)$ (uniformly) continuous and $f \in L^1$, then $\lim_{x\to\pm\infty}f(x)=0$?

I'm learning about measure theory and need help with the following questions: True or False (justify): $(1)$ If $f:\mathbb{R}\to[0, \infty)$ measurable and $f \in L^1$, then ...
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1answer
18 views

Algebra of Subsets with disjoint sets whose unions are equal

If $A$ is an algebra (of subsets of $\mathbb{R}$) and $E_1,E_2,...$ are elements of $A$. Show that there are dijoint sets $F_1,F_2,...$ elements of $A$ with $\cup E_n=\cup F_n$ from $n=1$ to infinity. ...
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1answer
28 views

Why a Collection of disjoint open sets in $\mathbb R^n$ has only countably many nonempty sets?

I am confused by this question. Why can't we just have a disjoint union of open sets of which every set is non empty?
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0answers
23 views

Measurability properties of functions

If f is measurable prove that any positive integral power of f is also measurable. Note: f is Lebesgue measurable. I wanted a proof from the definition of Lebesgue measurability of a function.
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1answer
16 views

Compute $\int_{B(0,2)} (xy^2+y^2z)d\mu(x,y,z)$ where $\mu$ is defined by an another integral

I have a measure $\mu$ defined as follow: for every a measurable set $E \subset \mathbb{R}^3$ $$\mu(E)=\int_{E \cap B(0, 1)} \sqrt{x^2+y^2+z^2}d\lambda_3(x,y,z)$$ where $\lambda_3$ is the Lebesgue ...
0
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1answer
22 views

Sequence of integrable functions that converge a.e but not their integral

I'm trying to find an example of a sequence of integrable functions $(f_n)_{n\in\mathbb{N}}$ such that $f_n\rightarrow 0$ a.e. (almost everywhere) but $\int f_n\nrightarrow 0$. Should be easy, but I ...
2
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2answers
38 views

Part of proof to show Lebesgue-lebesgue measurable

I want to prove the following: Suppose $E$ is a subset of $\Bbb R$, let $\gamma(E)=\{ (x,y)\in \Bbb R \times \Bbb R :x-y\in E\}$. If $E\in \Bbb B$ (Borel/Lebesgue measurable set), show that ...
2
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0answers
15 views

Approximate an integrable function using a simple function (Proving existance)

Let $f \in L^1(\mathbb{R})$, and let $\epsilon > 0$. Show that exists simple function $g=\sum_{k=1}^{n}c_k 1_{A_k}$, such that, $$\int_\mathbb{R} |f(x)-g(x)|dx \leq \epsilon$$,and such that $n \in ...
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0answers
33 views

Show that $\iint_{X \times Y}\varphi(x)k(x,y)\psi(y) d(\mu \times \nu)=\int_Y \Big[\int_X\varphi(x)k(x,y)d\mu \Big] \psi(y) d\nu$

Let $k(x,y)$ be a bounded Borel measurable function on $X \times Y$ and let $\mu$ and $\nu$ be Radon measure on $X$ and $Y$ i. Show that $\iint_{X \times Y}\varphi(x)k(x,y)\psi(y) d(\mu \times ...
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1answer
26 views

Inequality of Lebesgue integrals

Let $f,g\in\mathbb{L}(E)$. Suppose that $f\leq g$ and $A:=${$x\in E| f(x)<g(x)$}. Prove that $\int_{E}f<\int_{E}g$ if and only if $A$ has positive measure.
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1answer
20 views

Integrable functions in $\mathbb{R}$? [on hold]

Let $f\in\mathbb{L}(\mathbb{R})$ integrable. If $a>0$, prove that $f^{-1}((a,+\infty))$ has finite measure.
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0answers
56 views

Prove that Lebesgue measurable set is the union of a Borel measurable set and a set of Lebesgue measure zero

Let $A$ be a Lebesgue measurable subset of $\Bbb R$. 1) Show that there exists a Borel measurable subset $B$ of $\Bbb R$ such that $A\subseteq B$ and such that $l^*(B\setminus A)=0$. 2) ...
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0answers
20 views

how can I prove that? [on hold]

A step function is, by definition, a finite linear combination of characteristic functions of bounded intervals in R. Assume $f \in L^1( \Bbb R^1)$, and prove that there is a sequence $\{g_n\}$ of ...
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1answer
40 views

Regarding a Lebesgue measurable set

Let $A$ be a Lebesgue measurable set such that $m(A)>1$. Show that there exist $a,b\in A$ such that $a-b\in \mathbb Z$. Suppose this is not correct. Then for all $a,b\in A$ such that $a-b\notin ...
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0answers
21 views

Construction of a Borel subset $E$ of $\Bbb R$ such that both $E$ and $E^c$ has positive “density” everywhere.

Let $m$ be the Lebesgue measure on $\Bbb R$, then find $E$ Borel, such that for all $a<b, a,b\in\Bbb R$, $$m(E\cap(a,b))>0,\,m(E^c\cap(a,b))>0. $$ I couldn't find one. But after some ...
2
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1answer
14 views

A set with positive outer measure

Let $A$ be a subset of $\mathbb R$ such that $m^*(A)>0$. I want to show that there exist $x,y\in A$ such that $x-y\in \mathbb{R\setminus Q}$. Suppose for all $x,y\in A, x-y\in \mathbb Q$. We ...
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1answer
12 views

Closed set with finite measure

Assume $F\subset \mathbb R$ is closed and $\mu(F)<\infty$, where $\mu$ is the Lebesgue measure. Can we deduce that $F$ is bounded?
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1answer
21 views

About a measurable set

Let $A$ be a Lebesgue measurable set with $m(A)=8$. How to show that there exists a measurable set $B\subset A$ such that $m(B)=5$? I am not getting any idea how to proceed? Any hint will be ...
2
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0answers
35 views

f left continuous & strictly increasing; B Borel $\implies$ f(B) Borel (or at least Lebesgue Measurable)?

How's it going? In an attempt to use the Radon-Nykodym theorem to bulldoze through the admission of measures by bounded variation & monotonic functions (sidestepping all that Caratheodory ...
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0answers
14 views

In Egorov's Theorem, is almost everywhere same as point-wise?

I am studying about Egorov's Theorem. My teaching assistant said to me that Egorov's theorem is roughly like the following statement: Under two conditions which are $|E|\lt+\infty$ and ...
1
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1answer
31 views

General property regarding outer measure for a nested sequence of sets (measurable or not).

Let $\bigcap_{n=1}^\infty E_n=∅$ and if $\mu^*(E_n) <\infty$ and $E_{n+1} \subseteq E_n $ then $\lim\limits_{n\mapsto \infty} \mu^*(E_n) =0 $ even if each $E_n$ is a non-measurable set, where ...
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0answers
43 views

Prove or disprove a set function is a measure

Given the sequence of finite measures $\left\{t_n\right\}$ on a measurable space $(A, M)$ with $\sup_{n\in N}\ t_n(A)<\infty$. Let $u: M\rightarrow [0,\infty]$ be $u(E) = \sum_{n=1}^{\infty} ...
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10 views

Should The domain in Simple approximation theorem be measurable?

Here's Royden's version of The Simple Approximation Theorem, My question is do we really need domain $E$ be measurable? Or in other way, do we always define measurable function on a measurable ...
4
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1answer
36 views

Let $f: [0, 1] \to \mathbb{R}$ s.t $f(0)=f(1)=0$ then measure of $A = \{h \in [0, 1] \mid \exists x \text{ such that }f(x+h) =f(x)\} \geq 1/2$.

Let $f$ be continuous function from [0, 1] to $\mathbb{R}$ s.t $f(0)=f(1)=0$. Let $A = \{h \in [0, 1] \mid \exists x \text{ such that }f(x+h) =f(x)\}$. Show that set $A$ has Lebesgue measure $\geq ...
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1answer
40 views

Are every measure on $\mathbb{R}^{n}$ borel and/or regular?

I saw the above question in an exam paper and I am not sure how to even start. The question is true for the case of Lebesgue measure but I am not sure for arbitrary measures. I have tried looking at ...
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0answers
22 views

Show that step functions are dense in $L^1 (\Bbb R)$ [closed]

A step function is, by definition, a finite linear combination of characteristic functions of bounded intervals in $\Bbb R$. Assume $f \in L^1( \Bbb R)$, and prove that there is a sequence $\{g_n\}$ ...
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0answers
32 views

Proving $m(kE) = k\cdot m(E)$

Prove that $m(kE) = k\cdot m(E)$ given $k$ is a real number $k > 0$ $m$ is Lebesgue measure in $\mathbb{R}$ Here's my proof. Let $\{I_n\}$ be a collection of open intervals that cover $E$ $m(E) ...
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0answers
14 views

Example of a non-measurable set for which the outer measure is known [duplicate]

Do we know any set such that the set is not Lebesgue measurable but the outer measure of the set is known? If yes, then what is that set? If no, can we claim that all sets for which outer measure is ...
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1answer
25 views

Clean proof for showing $f^{-1}([a,\infty))$ measurable implies $f^{-1}([a,b))$ measurable

I wish to show that for $f:\mathbb{R} \to \mathbb{R}$, $f^{-1}([a,\infty))$ measurable implies $f^{-1}([a,b))$ measurable Looks fairly easy if $f^{-1}([a,b))$ is one piece. Suppose ...
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1answer
19 views

Are $L_\infty$ functions measurable/integrable?

Lemma 2.6 of "Ergodic Theory with a view towards Number Theory" (Einsiedler-Ward) involves: $$ \int f d\mu $$ where $f \in L^{\infty}$. Actually it is a calligraphic $L$ and I'd love if you would ...
3
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1answer
38 views

Deduce that $f=0 \operatorname{a.e.}$

Let $f:[a,b]\to \mathbb R$ be a measurable function .Then Prove that if $\int _c ^d f(x)\operatorname {dx}=0$ for all $a\le c <d\le b$ then $f=0 \operatorname{a.e.}$ My try: Let ...
0
votes
1answer
34 views

Show if $f(x)$ is measurable then $f(x+a)$ is also measurable

Prove or Disprove: If $f:\mathbb{R}\to\mathbb{R}$ then for any scalar $a$, the function $g_a :\mathbb{R}\to\mathbb{R}, g_a(x) = f(x+a)$ is measurable. I'm pretty sure this is true because ...
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0answers
16 views

Continuous function rational for every point, Cantor function

For Cantor function (https://en.wikipedia.org/wiki/Cantor_function), in my sense it is rational on every point. But it is continuous on [0,1], then such a function must be constant. What is the ...
2
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1answer
81 views

Proving Fatou type lemma

Let $f_1, f_2, \cdots$ and $f$ be nonnegative lebesgue integrable functions on $\mathbb{R}$ such that $$\lim_{n \to \infty}\int_{-\infty}^y f_n(x)dx = \int_{-\infty}^y f(x)dx \; \; \text{ for each ...
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0answers
12 views

Show that measure is regular

Let $\mathcal{B}$ denote the Borel-$\sigma$-algebra with respect to the Euclidean topology $\mathcal{T}$. Show that the measure $\lambda_{(0,1)}$ is regular. I start with outer ...
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0answers
23 views

Second part pf the exercise 10.J of the elements of integration and Lebesgue of bartle's book

Let (X,X,$\mu$) be the measure space on the natural numbers X=$\mathbb{N}$ with the counting measure defined on all subsets of X=$\mathbb{N}$. Let (Y,Y,$\nu$) be an arbitrary measure space. A ...
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0answers
18 views

finding the invariant measure of the map:$ f(x)=\frac {1}{1+x} $

Find the invariant measure of the map:$ f(x)=\frac {1}{1+x} $ I am not sure what exactly to do here. I'm pretty confused on the subject. I believe I have to find the location at where the area is ...
2
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0answers
23 views

Does $f(x,y) = (x^2-y^2,xy)$ send measurable set to measurable set?

I know that $f(x,y) = (x^2-y^2,xy)$ is not lipschitz. Maybe it would be locally lipschitz, then I think apply standard argument that for each bounded measure zero set is measure zero, and extend it to ...
2
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2answers
25 views

Proof outer measure satisfies monotonicity: $A \subseteq B \implies m^*(A) \leq m^*(B)$

Theorem: $$A \subseteq B \implies m^*(A) \leq m^*(B)$$ Proof Attempt: By definition, $m^*(B) = \inf\{\sum\limits_{k=1}^\infty |J_k||\{J_k\} \text{ is a cover of B }\}$, $m^*(A) = ...
0
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0answers
24 views

Given two finite measures $\mu$ and $\lambda$ on $R$, prove the following:

Given two finite measures µ and λ on R, show that (a) $\nu = λ + µ$ is a finite measure (b) $µ << \nu$ and $λ << \nu$ (c) Write $$\mu(E) = \int_E f d\nu$$ $$λ(E) = \int_E g d\nu$$ Let ...
1
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1answer
35 views

Equivalence Class of functions and properties examples

(1) A function $f:\mathbb{R}^d \rightarrow \mathbb{R}$ is continuous almost everywhere, if the set {x: $f$ is not continuous at x} is a null set (2) There exists a continuous function $g:\mathbb{R}^d ...
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0answers
18 views

Question on bounded function and Radon Nikodym Derivative.

Given $\lambda << \mu$ finite measures, show that for any bounded function $g$ and $A \subset [0,1]$ $$\int_A gd\lambda = \int_A ghd\mu$$ where $h$ is the Radon-Nikodyn derivative ...
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0answers
21 views

Finding conditions on functions for absolute continuity of measure

Let $\mu$ and $\lambda$ be finite measures of $[0,1]$ which are absolutely continuous with respect to Lebesgue measure: $\lambda << m$ and $\mu << m$ $$\lambda(A) = \int_Af(x)dx$$ ...
1
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1answer
51 views

Does length $[0,1]$ = length $(0,1)$?

So we know that the length of an interval $[a,b]$ is simply $b-a$ but does this hold if the interval is open? Or if one of the sides are open, like $(a,b]$ or $[a,b)$? Also, can I just confirm that ...
0
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1answer
25 views

Monotonic increasing and convergence in measure

If for each $n\in\mathbb{N}$, $f_n$ is monotonic increasing on [0,1] and $f_n\rightarrow f$ in measure, then $f_n\rightarrow f$ at every x at which f is continuous. I'm not sure whether this is right ...
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0answers
16 views

Relationship between Convergence in mean, convergence in measure and a.e. convergence

What is the relationship between convergence in mean under 1-norm (http://mathworld.wolfram.com/ConvergenceinMean.html), convergence in measure and a.e. convergence? I have shown that convergence in ...
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1answer
18 views

Exercise 10.J of The elements of integration and Lebesgue measure Bartle's book

The part of the problem is the next. Let (X,X,$\mu$) be the measure space on the natural numbers X=$\mathbb{N}$ with the counting measure defined on all subsets of X=$\mathbb{N}$. Let (Y,Y,$\nu$) be ...