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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).

45
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2answers
12k views

Lebesgue measurable but not Borel measurable

I'm trying to find a set which is Lebesgue measurable but not Borel measurable. So I was thinking of taking a Lebesgue set of measure zero and intersecting it with something so that the result is not ...
36
votes
2answers
3k views

Are most matrices diagonalizable?

More precisely, does the set of non-diagonalizable (over $\mathbb C$) matrices have Lebesgue measure zero in $\mathbb R^{n\times n}$ or $\mathbb C^{n\times n}$? Intuitively, I would think yes, since ...
35
votes
4answers
8k views

What does it mean for a set to have Lebesgue measure zero?

I am studying examples of sets with Lebesgue measure zero (e.g. the Cantor Set) but wanted an intuitive description of what this means rather than a formal definition. Thank you.
32
votes
5answers
1k views

What is wrong in this proof: That $\mathbb{R}$ has measure zero

Consider $\mathbb{Q}$ which is countable, we may enumerate $\mathbb{Q}=\{q_1, q_2, \dots\}$. For each rational number $q_k$, cover it by an open interval $I_k$ centered at $q_k$ with radius $\epsilon/...
29
votes
3answers
13k views

What is the difference between outer measure and Lebesgue measure?

What is the difference between outer measure and Lebesgue measure? We know that there are sets which are not Lebesgue measurable, whereas we know that outer measure is defined for any subset of $\...
28
votes
4answers
2k views

Apparent inconsistency of Lebesgue measure

Studying the Lebesgue measure on the line I've found the following argument which concludes that $m(\mathbb{R}) < +\infty$ (where $m$ denotes the Lebesgue measure on $\mathbb{R}$). Obviously it ...
26
votes
2answers
2k views

Lebesgue density strictly between 0 and 1

I am having trouble with the following problem: Let $A\subseteq \mathbb{R}$ be measurable, with $\mu(A)>0$ and $\mu(\mathbb{R}\backslash A)>0$. Then how do I show that there exists $x\in \...
23
votes
2answers
6k views

What's the relationship between a measure space and a metric space?

Definition of Measurable Space: An ordered pair $(\Omega, \mathcal{F})$ is a measurable space if $\mathcal{F}$ is a $\sigma$-algebra on $\Omega$. Definition of Measure: Let $(\Omega, F)$ be a ...
20
votes
7answers
599 views

When does $\lim_{n\to\infty}f(x+\frac{1}{n})=f(x)$ a.e. fail

We know that if $f\in L^1(\mathbb{R})$, then $\|f(\cdot+1/n)-f(\cdot)\|_{L^1}\to 0$ as $n\to \infty$, which implies that there exists a subsequence $f_{n_k}=f(x+\frac{1}{n_k})$ such that $f_{n_k}\to f$...
19
votes
1answer
3k views

Vitali set of outer-measure exactly $1$.

I know that for any $\varepsilon\in (0,1]$ we can find a non-measurable subset (w.r.t Lebesgue measure) of $[0,1]$ so that its outer-measure equals exactly $\varepsilon$. It is done basicly with the ...
18
votes
4answers
3k views

Prove that every Lebesgue measurable function is equal almost everywhere to a Borel measurable function

Suppose $(\mathbb{R},\Sigma(m),m)$ is our measure space, where $m$ is Lebesgue measure. Also, suppose $f : \mathbb{R} \to [-\infty, \infty]$ is a Lebesgue measurable function. The problem: Prove ...
18
votes
1answer
2k views

Why is Lebesgue-Stieltjes a generalization of Riemann-Stieltjes? Moreover, is there an example where Lebesgue-Stieltjes is useful

I certainly have a question, but i don't know what the best title should be. Please edit the title if there is a better one :) And I believe, to get a better answer, it would be good to explain ...
17
votes
1answer
2k views

Construct a Borel set on R such that it intersect every open interval with non-zero non-“full” measure

This is from problem $8$, Chapter II of Rudin's Real and Complex Analysis. The problem asks for a Borel set $M$ on $R$, such that for any interval $I$, $M \cap I$ has measure greater than $0$ and ...
16
votes
1answer
491 views

If $f$ is Lebesgue integrable on $[0,2]$ and $\int_E fdx=0$ for all measurable set E such that $m(E)=\pi/2$. Prove or disprove that $f=0$ a.e.

Let $f$ be a Lebesgue integrable function on $[0,2]$. If $\int_E fdx=0$ for all measurable set $E$, such that $m(E)=\pi/2$. Is $f=0$ a.e. Prove or disprove I could not figure out anything. Can a ...
15
votes
2answers
1k views

Does the graph of a measurable function always have zero measure?

Question: Let $(X,\mathscr{M},\mu)$ be a measure space and $f\colon X \to [0,+\infty[$ be a measurable function. If ${\rm gr}(f) \doteq \{ (x,f(x)) \mid x \in X \}$, then does it always satisfy $$(\mu ...
15
votes
1answer
370 views

Subset of plane with measure $1$ in all lines

Is it possible for there to be an $X\subset\Bbb R^2$ such that for all lines $\ell$ in the plane, we have that $X\cap\ell$ has measure $1$ in $\ell$? I don't know enough measure theory to see how to ...
14
votes
2answers
793 views

Are dense subsets almost nothing or almost everything?

Dense subsets of $[0,1]$ I know have Lebesgue measure $0$ or $1$, but, is there any dense, uniform subset of $[0,1]$ with meausre $1/2$? What I mean with uniform: a subset $A$ of $[0,1]$ is uniform ...
14
votes
4answers
3k views

Prove Borel Measurable Set A with the following property has measure 0.

This question is exercise 4.10 of Richard F. Bass's Real Analysis for Graduate Students, 2nd edition. Let $\epsilon \in (0,1)$, let $m$ be Lebesgue measure, and suppose $A$ is Borel Measurable subset ...
14
votes
3answers
812 views

If $f'(x)>0$ on $E$ , where $m(E)>0,$ then $m(f(E))>0$

Let $f:\mathbb R\to \mathbb R,$ and suppose $f$ is differentiable at every point of a measurable $E\subset \mathbb R,$ with $f'>0$ on $E$. Suppose also that $m(E)>0$ (where $m$ is Lebesgue ...
13
votes
3answers
177 views

Prove that $\vert E \vert=0$ if $\frac{x+y}{2}\notin E$ for $x,y \in E$

Let $E\subset\mathbb{R}$ measurable such that if $x\neq y, x,y\in E$ then $\frac{x+y}{2}\notin E$. Prove that $\vert E \vert=0$. Any idea? Thanks!
13
votes
1answer
2k views

Completion of borel sigma algebra with respect to Lebesgue measure

There are two ways of extending the Borel $\sigma$-algebra on $\mathbb{R}^n$, $\mathcal{B}(\mathbb{R}^n)$, with respect to Lebesgue measure $\lambda$. The completion $\mathcal{L}(\mathbb{R}^n)$ ...
13
votes
1answer
244 views

Number of flips to get to a Set of Positive Lebesgue Measure

A consequence of Exercise 1.1.19 on page 13 of Stroock's "Probability Theory: An Analytic View" is that if a set $E\subset[0,1)$ has positive (Lebesgue) measure, then for almost every $x\in[0,1)$, a ...
12
votes
2answers
3k views

Nowhere dense subsets of $[0,1]$ with positive measure other than fat Cantor sets

This is my first time on the board, so forgive me if I've posted incorrectly. In any case, I think my title is self-explanatory: the only examples I've encountered for nowhere dense subsets of $[0,1]$ ...
12
votes
1answer
4k views

Are all measure zero sets measurable?

Definition of Lebesgue Outer Measure: Given a set $E$ of $\mathbb R$, we define the Lebesgue Outer Measure of $E$ by, $$m^*(E) = \inf \left\{\sum_{n=1}^{+\infty} \ell(I_n): E \subset \bigcup_{n=1}^{+\...
12
votes
3answers
545 views

measurability with zero measure

Let $f : [0,1] \rightarrow \Bbb R$ is arbitrary function , and $E \subset \{ x \in [0,1] | f'(x)$ exists$\}$. How to prove this statement: If $E$ is measurable with zero measure then $f(E)$ is ...
12
votes
1answer
2k views

The integral of a characteristic function with respect to a product measure.

Problem: Let $ (X,\mathcal{A},\mu) $ and $ (Y,\mathcal{B},\nu) $ be measure spaces, where $ X = Y $ is the interval $ [0,1] $, $ \mathcal{A} = \mathcal{B} $ is the collection of Borel ...
12
votes
1answer
267 views

Non-invertible measure preserving transformations of $\mathbb{R}^n$

I am looking for particular examples of measure-preserving transformations of $\mathbb{R}^n$ (with Lebesgue measure) to get a better idea of how they behave. A large family of such transformations ...
12
votes
1answer
255 views

Measure of set where holomorphic function is large

Suppose that $f:\mathbb{C}\rightarrow \mathbb{C}$ is a non-constant entire function. By Liouville's theorem, we know that $f$ must take on arbitrarily large values. However Liouville doesn't say ...
12
votes
2answers
683 views

Topology of convergence in measure

Currently I am doing some measure theory (on $X=[0,1]$ with the Borel-Sigma algebra and the Lebesgue measure), and I am looking at sets $A \subset L^p$, such that for all $q \in (0,p)$, the topologies ...
11
votes
4answers
817 views

Why “countability” in definition of Lebesgue measures?

According to Wikipedia, the definition of the Lebesgue outer measure of a set $E$ is as follows: $$ \lambda^*(E) = \operatorname{inf} \left\{\sum_{k=1}^\infty l(I_k) : {(I_k)_{k \in \mathbb N}} \text{...
11
votes
2answers
4k views

Why is the Monotone Convergence Theorem restricted to a nonnegative function sequence?

Monotone Convergence Theorem for general measure: Let $(X,\Sigma,\mu)$ be a measure space. Let $f_1, f_2, ...$ be a pointwise non-decreasing sequence of $[0, \infty]$-valued $\Sigma-$measurable ...
11
votes
3answers
6k views

A function that is Lebesgue integrable but not measurable (not absurd obviously)

I think: A function $f$, as long as it is measurable, though Lebesgue integrable or not, always has Lebesgue integral on any domain $E$. However Royden & Fitzpatrick’s book "Real Analysis" (4th ...
11
votes
3answers
2k views

Prove that $f$ is integrable if and only if $\sum^\infty_{n=1} \mu(\{x \in X : f(x) \ge n\}) < \infty$

Problem statement: Suppose that $\mu$ is a finite measure. Prove that a measurable, non-negative function $f$ is integrable if and only if $\sum^\infty_{n=1} \mu(\{x \in X : f(x) \ge n\}) < \infty$....
11
votes
2answers
1k views

Intuitive Explanation of Why the Power Set of $\mathbb{R}$ is “too big” for the Lebesgue Measure?

I've been working with the construction of measures for a little bit, and I understand that in order for the Lebesgue measure to be an official measure on $\mathbb{R}$, we need to restrict it to a ...
11
votes
1answer
2k views

The completion of the Borel $\sigma$-algebra the same as the completion of the Lebesgue outer measure?

My study group and I were discussing this question today. We can construct the Lebesgue measure using Caratheodory's extension theorem in the usual way: Given the function $F(x) = x$, we can ...
11
votes
0answers
119 views

Is there a Lebesgue measurable subset $A \subset R$ such that for every interval $(a,b)$ we have $0 < \lambda(A\cap(a,b))< (b-a)$ [duplicate]

Question: Is there a Lebesgue measurable subset $A \subset R$ such that for every interval $(a,b)$ we have $$0 < \lambda(A\cap(a,b))< (b-a)$$ It looks like the answer is no, I'm trying to use ...
11
votes
0answers
207 views

Minimum area contained between measurable set and translate by $\lambda$: A strengthening of 2018 USA TSTST #9

Question Given $\lambda\in\mathbb{R}^+$, what is the smallest possible $c$ for which, given any measurable region $\mathcal{P}$ in the plane with measure $1$, there always exists a vector $\mathbf{v}$...
10
votes
3answers
272 views

Two inequalities about using Fatou Lemma

1) Let $\{f_n\}$ be a sequence of nonnegative measurable functions of $\mathbb R$ that converges pointwise on $\mathbb R$ to $f$ integrable. Show that $$\int_{\mathbb R} f = \lim_{n\to \infty}\int_{\...
10
votes
3answers
285 views

How is area defined?

Thinking about area in the context of the Lebesgue measure, I have an intuitive understanding of how area is constructed in $\mathbb{R}^2$: define all rectangles to have the area $length \times width$...
10
votes
1answer
168 views

Convergence of a series of translations of a Lebesgue integrable function

Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be a Lebesgue integrable function. Prove that $$\sum_{n=1}^{\infty} \frac{f(x-\sqrt{n})}{\sqrt{n}}$$ converges almost for every $x \in \mathbb{R}$. My ...
10
votes
1answer
2k views

A rigorous meaning of “induced measure”?

In my readings I often come across terms like "induced measure" or "induced Lebesgue measure". For example: $$\int_{\mathbb{B}^n}u\frac{\partial v}{\partial x_j}\;dx = \int_{\mathbb{S}^{n-1}}uv\...
10
votes
1answer
268 views

Intuition for $N(\mu, \sigma^2)$ in terms of its infinite expansion

To gain deeper insight to the Poisson and exponential random variables, I found that I could derive the random variables as follows: I consider an experiment which consists of a continuum of trials ...
10
votes
0answers
977 views

probability measures vs. probability distributions vs. measure of probability density

I am learning probability theory right now and am confused about some basic concepts. I have a few questions and am wondering if you can also check if the following is correct: Suppose we have a ...
9
votes
3answers
2k views

Is the measure of the sum equal to the sum of the measures?

Let $A,B$ be subsets in $\mathbb{R}$. Is it true that $$m(A+B)=m(A)+m(B)?$$ Provided that the sum is measurable. I think it should not be true, but could not find a counterexample.
9
votes
3answers
846 views

Is a function necessarily measurable, given that all of its level sets are measurable?

Let $f: \mathbb{R}^n \rightarrow \mathbb{R}$ be a function such that the set $$T_{\alpha} \equiv \{ x \in \mathbb{R}^n : f(x) = \alpha\}$$ is measurable $\forall \alpha \in \mathbb{R}$. Is $f$ ...
9
votes
3answers
4k views

Lebesgue density theorem in the line

Suppose $A \subseteq \mathbb{R} $, $m(A) > 0 $. Then for almost all $x \in A $ we have $$ \lim_{\epsilon \to 0^+ } \frac{ m(A \cap (x - \epsilon, x + \epsilon))}{2 \epsilon} = 1.$$ Can someone ...
9
votes
4answers
660 views

Can we find uncountably many disjoint dense measurable uncountable subsets of $[0,1]$?

Can we find uncountably many disjoint dense measurable uncountable subsets of $[0,1]$? Obviously we may as well assume all the subsets have measure $0$. If I didn't specify the subsets were ...
9
votes
4answers
476 views

Elementary Lebesgue measure problem

Suppose $E_1,\cdots,E_n\subset [0,1]$ are Borel sets such that $\sum_{i=1}^n\mu(E_i)>n-1$, in which $\mu$ denotes Lebesgue measure. Prove that $\cap_{i=1}^nE_i$ is nonempty. My attempts included ...
9
votes
3answers
8k views

Why is the outer measure of the set of irrational numbers in the interval [0,1] equal to 1?

Just learned Lebesgue outer measure from Royden's Real Analysis. Let me give my proof. First, let $A$ be the set of irrational numbers in [0,1]. So $A\subset [0,1]\Rightarrow m^*(A)\le m^*([0,1])=1$....
9
votes
2answers
304 views

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 ...