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).

learn more… | top users | synonyms

1
vote
2answers
61 views

Do Riemann and Lebesgue integrals always agree?

I know that on a closed bounded interval, say $[a,b]$ in $R^1$, if a function is Riemann integral, then it is Lebesgue integrable, and the values of those two integrals are the same. But, is this ...
3
votes
1answer
26 views

Are measurable sets closed under projections?

For the following, let us assume that large enough sets to carry the arguments through do exist, i.e. that there are supercompact cardinals or whatever is sufficient. I know that all projective ...
4
votes
1answer
29 views

Lebesgue measure without choice

From this question and this question (and their answers) I gather that it is consistent with ZF without The Axiom of Choice to assume that there exist countable sets $A_n$, $n\in \mathbb N$, such that ...
0
votes
1answer
38 views

Why does a Borel measurable function imply its Lebesgue measure?

Borel measurable defined as: $f: D ->\mathbb R$ is Borel measurable if $D$ is a Borel set and if, for each real $a$, the set {$x∈D: f(x) > a$} is a Borel set. Definition of Lebesgue measurable ...
1
vote
1answer
20 views

Given $f\in L^1(\mathbb{R})$ with $||f||_1 < \infty$, is it true that $\int_{\mathbb{R}} ||f||_1 - f(x) \, dx = 0$?

According to my intuition so far, the answer should be yes, hinging very important on the assumption that $||f||_1 < \infty$. To speak very roughly, if the $L^1$ norm of $f$ is finite, it seems ...
0
votes
0answers
12 views

show that $f_n\to 0$ (lebesgue measure), is not convergent in any point of $x \in [0,1]$? [on hold]

Prove that $f_n\to 0$ (lebesgue measure), is not convergent in any point of $x \in [0,1]$? Hint: let $f_n$ be the characteristic function of n-th interval in: ...
1
vote
3answers
64 views

Compact subset of $\mathbb R$ whose Lebesgue measure is non-zero

Let $\mathbb R$ be the field of real numbers, $\mu$ the Lebesgue measure on it. Let $K$ be a compact subset of $\mathbb R$. Is the following assertion true? If $\mu(K) \gt 0$, then the interior ...
1
vote
0answers
25 views

Defining Lebesgue measure on a subspace of $\mathbb{R}^n$

Let $\bar{w}_1,.., \bar{w}_k$ be linearly independent vectors in $\mathbb{R}^n$. Let $W$ be the subspace spanned by these $\bar{w}_i$'s. I know how the Lebesgue measure is defined on $\mathbb{R}^n$. ...
0
votes
3answers
58 views

Lebesgue Measure of $\mathbb{R} \times \{0\}$

I know this is probably obvious, and I know the answer is that it is (Lebesgue) measure zero, but I'm having a hard time wrapping my head around it. Looking for an intuitive explanation. Question: ...
0
votes
1answer
40 views

What are the hypotheses in Levi's monotone convergence theorem?

Today I read monotone convergence theorem , dominated convergence theorem and fatou's lemma And I need some help We know the dominated convergence theorem in Measure theory In its proof we ...
1
vote
1answer
58 views

Is a countable intersection of open sets in $\mathbb R$ Lebesgue measurable?

If the answer is yes, how to prove that? Otherwise how to find a counterexample? Update: I've figured out the tricks inside. A countable intersection of open sets in $\mathbb R$ is equivalent to a ...
2
votes
1answer
54 views

Prove Y = X given $Y = E[X|\mathscr{G}] $ and $EY^2 = EX^2$

Prove Y = X, given $Y = E[X|\mathscr{G}] $ and $EY^2 = EX^2$ Attempt: Suppose $Y = E[X|\mathscr{G}] $. Then $E[X|\mathscr{G}] $ is $\mathscr{G}$-measureable. For every A $\in \mathscr{G}$: ...
1
vote
0answers
42 views

a real analysis question,I need to prove whether two sequences are equidistributed or not,really need some help!

For each subset $S\subset [0,1]$ write $X_{s}$ for the "indicator function" as $X_{s}(t)=1$ when $t\in S$ and $X_{s}(t)=0 $ when $t\notin S$. a sequence $\{x_{n}\}$ in [0,1] is called ...
0
votes
1answer
14 views

Finite measure space & sigma-finite measure space

A measure space $(X, \Sigma, \mu)$ is finite if $\mu(X)<\infty$. It is equivalent to saying that $(X, \Sigma, \mu)$ is finite if $\mu(E)<\infty$ for all $E \in \Sigma$ A measure space $(X, ...
1
vote
2answers
54 views

Is a subspace of functions that essentially depend only on one variable closed?

Let $S$ be the subspace $$\left\{f\in L^p( I^2)|\exists g\in L^p( I), f(x,y)=g(x), \mbox{a.e. } (x,y)\in I^2\right\}.$$ Is $S$ closed under the $L^p$ norm? I think the first step would be to ...
0
votes
1answer
17 views

Outer Measure on a Probability Space is 1 iff its complement is null?

I am trying to prove the following theorem, which I feel should be true, but am not sure how to go about it. Suppose we are in a probability space $(\Omega, \mathcal{F}, P)$ and we define for any ...
-1
votes
0answers
23 views

Disjoint simple functions dense in $L^2(I^2, R^2)$

Suppose $$S=\{f\in L^2(I^2,R^2)| \exists h_1(x), f_1(x,y)=h_1(x), a.e. (x,y)\in I^2, \\ \exists h_2(x), f_2(x,y)=h_2(y), a.e. (x,y)\in I^2\}$$ For any $f=(f_1,f_2)\in L^2(I^2, R^2)$ and any ...
0
votes
1answer
34 views

Congruent Sub-Intervals with Reimann-Integrable Functions

Let $f:[a,b]\to\Bbb R$ be a Riemann-integrable function. Prove that for each $\sigma\gt0$ there exists a partition $\mathcal P$ of $[a,b]$ into congruent sub-intervals(that is, $x_{j}=a+{j(b-a)\over ...
-2
votes
0answers
28 views

a question about real analysis,I need to know whether these two sequences are equidistributed. [closed]

For each subset $S\subset [0,1]$ write $X_{s}$ for the "indicator function" as $X_{s}(t)=1$ when $t\in S$ and $X_{s}(t)=0 $ when $t\notin S$. a sequence $\{x_{n}\}$ in [0,1] is called ...
1
vote
3answers
300 views

What's the Lebesgue measure of this set?

The following citation is from Folland's Real Analysis. Let $m$ denote the Lebesgue measure on $\mathbb{R}$ and $\{ r_j \}$ be an enumeration of the rational numbers in $[0,1]$, and given $E > ...
1
vote
0answers
16 views

Optimal $A\in \Sigma$ that maximizes an objective

Let $([0,1],\Sigma, \lambda)$ be a probability space. For any given $B\in \Sigma$, $K\in [0,1]$ and $f\in L^2(\lambda)$ with $f(x)\in[0,1]$ for all $x $, $$\max_{A\in \Sigma}\int_A f(x) d\lambda(x)- ...
2
votes
0answers
43 views

What's the relationship between continuity property of Lebesgue measure and continuity on a metric space?

This is a topic from Lebesgue measure in $\textit {Carothers' Real Analysis}$: I know how to prove Theorem 16.23. However, I can not figure out why he names this property as continuity? Besides ...
2
votes
1answer
48 views

a question about the evaluation of integral [duplicate]

Let $\alpha:[0,1] \to R$ be the Cantor function. Evaluate $$\int_{0}^{1}xd\alpha $$and $$\int_{0}^{1}x^2d\alpha.$$ I know that the Cantor function is continuous and monotone increasing, how can I ...
1
vote
0answers
51 views

Puzzles in a proof

From a previous link in MSE: Prove the set of which sin(nx) converges has Lebesgue measure zero (from Baby Rudin Chapter 11), the question states Suppose that $\{n_k\}$ is an increasing sequence ...
1
vote
2answers
96 views

Prove $f_n\to f$ on $[a,b]\implies \int_a^b|f_n-f|\to 0$

Suppose $f,f_n$ are measurable and uniformly bounded on $[a,b]$. Prove $f_n\to f$ on $[a,b]\implies \int_a^b|f_n-f|\to 0$ Attempt: We note that since $f$ and $f_n$ are bounded and are ...
0
votes
0answers
14 views

Linear functional on bounded functions in $\mathbb R$ that is translation invariant

This should be a simple problem but I can't figure it out. I'm attempting to construct a finitely additive measure on $\mathbb R$ which equals the Lebesgue measure on Lebesgue measurable sets, ...
4
votes
1answer
36 views

measure of a set invariant by rational translation

Say that a measurable subset $A$ of $[0,1]$ is ${\mathbb Q}$-stable if $a+q\in A$ whenever $a\in A,q\in{\mathbb Q}$ and $a+q\in [0,1]$. Obviously, $\emptyset$ is ${\mathbb Q}$-stable and has measure ...
0
votes
1answer
43 views

Any positive measure subset of $\mathbb R$ contains a positive measure Cantor set

A question asks to show any positive measure subset of $\mathbb R$ contains a positive measure Cantor set. How to start with this? I have been staring on this for a while, but can not come up with any ...
0
votes
0answers
24 views

Lebegue measure of the union of measurable sets

Let us call a subset $E$ of $[0,1]$ Lebesgue measurable if $\lambda^*(E) + \lambda^*([0,1]\setminus E) = 1$, where $\lambda^*$ is the outer measure. How can we derive from this definition the fact ...
5
votes
0answers
29 views

Showing that $f_n\to f$ a.e. $\implies f_n\to f$ almost uniformly where $|f_n|\leq g\in L_1$ [duplicate]

The following problem is from Carothers' Real Analysis: Suppose $f_n$ is a measurable sequence of functions such that $|f_n|\leq g\in L_1$ for all $n$. Prove that $f_n\to f$ almost everywhere ...
2
votes
1answer
37 views

A basic question regarding Lebesgue's density theorem

Here is the question from Pugh's Real Mathematical Analysis: My answer to $b)$ is that for a closed square, points on corner has density $1/4$, while on the sides the density is $1/2$. But how to ...
0
votes
1answer
19 views

Cavalieri's Principle in measure theory

The first part of Cavalieri's principle (in measure theory) states if $E$ is measurable, then almost every slice $E_x$ of $E$ is measurable. Here, it uses "almost every", so what is an example where ...
0
votes
0answers
22 views

Lebesgue Measurable By Alternative definition of Measure

Prove that any compact set $K$ in $R^{n}$ is Lebesgue measurable and $m(K) < \infty$ Actually the proof of this is given in Stein and Shakarchi's book on Real Analysis (Page 38, Property 4) where ...
1
vote
0answers
35 views

If 2-dimensional Lebesgue measure is one then $S_x$ has 1-dimensional measure of one

I am working on a problem$^{(1)}$ like this: Suppose $A$ is a Lebesgue measurable subset of [0, 1]$^2$ with 2-dimensional Lebesgue measure $m_2 (A) = 1$. Show that for almost every $x \in [0, 1]$ ...
1
vote
1answer
32 views

A class of functions dense in $L^2$

Suppose $f\in L^2([0,1],\Sigma,\mu)$. Is the class of all $$f=\sum_{i=1}^n \alpha_i (\chi_{A_i}-\chi_{[0,1]/A_i} )$$$A_i\in \Sigma$ to be dense in $L^2([0,1],\Sigma,\mu)$? Thanks.
1
vote
0answers
38 views

Absolutely continuous iff continuous of bounded variation

I have the following problem (taken from q1 p341 of Kolmogorov and Fomin's Introductory Real Analysis), which I am struggling to prove completely. I think I know how to show the only if part, but not ...
1
vote
0answers
27 views

Regularity of Special Measures

(1) Show that the counting measure on $\Bbb Z$ with the induced metric from $\Bbb R$ is regular. (2) Show that the delta measure with respect to a point $x_0$ on any metric space is regular. What I ...
0
votes
1answer
28 views

$f_n$ converge to $f$ almost everywhere (measure theory)

If $\mu(\{w \in A: |f_n(w) - f(w)| > \epsilon\,\, \text{for infinitely many many $n$}\}) = 0$ for each $\epsilon > 0$, then $f_n \rightarrow f$ $\mu$-a.e. I can see how to solve this problem ...
0
votes
1answer
41 views

$f_n$ converges to $f$ in measure iff $f_n-f$ converges to $0$ in measure.

Prove that $f_n$ converges to $f$ in measure iff $f_n-f$ converges to $0$ in measure. What I know: $(f_n)$ is said to converge in measure to $f$ if: $$m\{|f_n-f|\geq \epsilon\}<\epsilon ...
2
votes
1answer
49 views

Let $g$ be a non-negative measurable function. Show $\int g(x)^p d\mu = \int_0^\infty p t^{p-1} m_g(t) dt$.

Let $g$ be a non-negative measurable function. For $1 \leq p < \infty$, show that $$\int g(x)^p d\mu = \int_0^\infty p t^{p-1} m_g(t) dt$$ where $\mu$ is the Lebesgue measure and we are ...
5
votes
1answer
92 views

Complete example of haar measure on compact groups like $GL(n,R)$

I am currently reading the proof of existence of haar measure, but I learn better mostly by examples so I would like examples of explicit computation of haar measure mainly on any $Gl(n,R)$ or any lie ...
0
votes
0answers
45 views

Simple proof check---Exercise from Tao's real analysis notes, Jordan measure

It is a rather simple question, I just want to make sure if my way of doing it works. So the exercise is to proof the monotonicity of Jordon measure (http://en.wikipedia.org/wiki/Jordan_measure). If ...
1
vote
0answers
15 views

Subclasses of simple functions dense in $L^2$

Q1. Consider $f\in L^2([0,1], R)$ with $ ||f||^2=\int f(x)^2d\mu(x)$ Consdier a subclass of simple functions $f= \sum_{i=1}^n a_i \chi_{A_i}$ where $A_i\in \Sigma$ (on $[0,1]$) and ...
1
vote
0answers
23 views

Weird behavior of Non-Lebesgue measurable subset of the Smith–Volterra–Cantor set and pointwise convergence of a sequence of simple functions

I came across this (seemingly?) weird behavior of a sequence of simple functions: Let $E$ be the Smith–Volterra–Cantor set and $m: \operatorname{Leb}(\Bbb{R}) \to [0, \infty]$ the Lebesgue-measure. ...
0
votes
0answers
25 views

$j$-volume of $j$ dimensional parallelepiped inside $\mathbb{R}^n$

Let $v_1, ..., v_j \in \mathbb{R}^n$ be linearly independent. Let $V = \mathbb{R}v_1 + ... + \mathbb{R} v_j$ be a subspace of $\mathbb{R}^n$ and $\Gamma = \mathbb{Z}v_1 + ... + \mathbb{Z} v_j$ a ...
2
votes
1answer
24 views

The irrational rotation is ergodic. The proof should use the idea of density point.

Consider $f_{\alpha}:S^{1}\rightarrow S^{1}$ the rotation of unit circle of angle $2\pi\alpha$, and let $\mu$ the Lebesgue measure in $S^{1}$. Let $\alpha$ irrational, show that $\left(f,\mu\right)$ ...
4
votes
1answer
50 views

approximation of measurable functions

Hi we know we can approximate measurable function by simple function however can we increase the conditions such that we can approximate by at most countable functions that is there exists sequence ...
0
votes
0answers
14 views

definition for a function to be measurable

Hi I don't completely understand the following definition : Let A $\in$ F be nonempty , and let f : A $\rightarrow$ R denote a function. We will say that f is F/B*-measurable iff $f^{-1}(B)$ $\in$ F ...
2
votes
2answers
35 views

Lebesgue outer Measure of a face of rectangle in $\Bbb R^{n}$

Show that the outer measure of a face $I_1 \times \dots \times I_{i-1} \times \{a\} \times I_{i+1} \times \dots \times I_n$ of a rectangle $I_1 \times \dots \times I_n \subset \Bbb R^{n}$ is zero. ...
0
votes
0answers
22 views

Lebesgue nonmeasurable sets

Using the continuum hypothesis, prove that there exists a Lebesgue nonmeasurable subset $E$ of $\mathbb{R}^2$ such that $E$ intersects every horizontal or vertical line in exactly one point.