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).
7,412
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On the role of PDFs in the determination of impossible events
Let's consider a random variable $X$ distributed according to a PDF $p(x):\mathbb{R}\mapsto \mathbb{R}_{\geq 0}$. Is it true to say that for any $x_0:p(x_0)=0$ the event $X=x_0$ is impossible?
By ...
- 501
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Find all $c\in[3,\infty)$ such that there exists a measure space $(X,S,\mu)$ with $\{\mu(E) : E \in S\} = [0,1]\cup[3,c]$.
I believe that the values of c are 4 and infinity, but I did not understand the explanations you have presented on this site regarding the answer. Do you have better details?
-5
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Any set of $R^d$ is $G/delta$ or $F/sigma$ [closed]
is this true? We have to try it and we don't know where to start.
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1
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Exercise 8 Chapter 6 Rudin Real and Complex Analysis
I believe I have solved this problem, as it seems relatively straightforward but I am not sure whether the solution has a more 'neat' form in which it can be stated since the conclusion I reached ...
- 198
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Yet another property of the Cantor function; a bound on its variation?
Consider the Cantor function $f:[0,1]\to [0,1]$ and an element of the Cantor set $x\in C$, we want to prove that
$$f(x+3r)-f(x-3r)\leq 4 (f(x+r)-f(x-r))$$ for all $r>0$ such that the formula makes ...
- 838
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1
answer
44
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Change of measure for measures on different spaces
For two measures $\nu, \mu$ that are defined on the same measure space with $\nu = \int f d\mu$, it is a well known result, mostly used in the context of the radon nikodym theorem, that for an ...
- 333
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1
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$\lim_{t \to 0} \frac{1}{t^{1/q}}\int_{[0,t]}|f|dm=0$ where $f \in L^{p}$ and $q=p^*$
Consider the measure space $([0,1], \mathcal {B}_{[0,1]}, m)$ where $m$ is Lebesgue measure, and let $f \in L^{p}(m)$ for some $p>1$.
Let $q=p^*$ (that is, $1/p+1/q=1$).
Prove or disprove:
$$\lim_{...
- 127
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Would We have the preimage of at least one interval taken out from $[0;1]$, inside the any ball taken out from $X$?
This is the theorem:
Let $X$ be separable metric space endowed with non-atomic measure such that $\mu X = 1$.
Using this theorem We can establish isomorphism between $X$ and $[0;1]$.
Denote this ...
- 906
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1
answer
68
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Ambiguity in the definition of a simple function
I have seen the definition for a simple function, $\phi:X \to \mathbb{R}$ where $$\phi(x) = \sum_{i=1}^k c_i \chi_{A_i}(x)$$
This seems silly in my head but I was wondering if we could have for ...
- 57
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1
answer
33
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Measures which are absolutely continuous wrt to the Lebesgue measure
Let $dx$ denote the Lebesgue measure on $[0,1]$. Are there any measures $\mu$ on $[0,1]$ which are a.c. with respect to $dx$ that have support $[0,1]$, but are not equivalent to the Lebesgue measure, ...
- 186
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1
answer
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Lebesgue spaces $\mathscr{L}^{p}$ for $p<1$ [duplicate]
I have this following task for Lebesgue spaces $\mathscr{L}^{p}$ with $p<1$:
For $p\in(0,1)$ give an example of a measure space $(X,\mathscr{A},\mu)$ and $f,g\in\mathscr{L}^{p}(X,\mathscr{A},\mu;\...
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How to calculate the limits of the integral and the outward vector? How to make change of variables in this proof?
I would like to see the calculations behind this proof I found here:
Proof of polar coordinates theorem in Evans' PDE Book
That is, I am struggling to understand the following:
How do we ...
- 337
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25
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Locally essentially bounded function is locally integrable
This may be an obvious question, but I'm still doubting if my reasoning is ok.
There is a result that says the following:
If $1 \leq p \leq s \leq \infty$, and $|\Omega| < \infty$, ($\Omega$ is an ...
- 361
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Swapping integral and limit [closed]
Le $\left(f_k\right)_k$ be a sequence of nonnegative $\mathcal{L}^1$-measurable function on $\mathbb{R}$ s.t. $f_k \rightarrow f, \mathcal{L}^1$ almost everywhere. Then $\lim_{k \rightarrow \infty} \...
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1
answer
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Prove that $ \int^t_0X_s dA_s$ is progressively measurable.
Let $(\Omega, \mathcal F_\infty, \mathcal F= (\mathcal F_t)_{t\geq 0})$ be a filtered probability space, let $X = (X_t)_{t\geq 0}$ be a progressively measurable process and $A= (A_t)_{t\geq 0}$ be a ...
2
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+50
Transforming Radon Nikodym derivatives
I am currently confused by the (abuse of?) notation regarding the Radon Nikodym derivative in many proofs.
What I am currently struggling with in particular is a proof of the classical information ...
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answer
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Prove that $ \lim_{p\rightarrow 0^+}(\frac{1}{|E|}\int_E f^p)^{\frac{1}{p}}=\exp(\frac{1}{|E|}\int_E \log f) $
I ask this question two days ago, but no one answer or comment. Thus, I reedit my question and edit more about mine thought. Hope someone can help me prove that. Here the question:
Let $E$ be a ...
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23
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Existence of a continuous function that satisfies a constraint: $\|f - f_{\epsilon}\|_{1} \leq \epsilon$ with $f(t) \in A$ a.e. $t \in [0,1]$.
Given a measurable function $f \in L^{\infty}([0,1],\mathbb{R}^d)$, we know that for all $\epsilon > 0$ there exists a continuous function $f_{\epsilon}$ such that
$$ \|f - f_{\epsilon}\|_{L^1} \...
- 1,375
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41
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Lebesgue measure on $\mathbb{S}^n$ in practice
I want to compute $\int_{\mathbb{S}^n} f(y) d\mu (y)$ with $n=2$, $f$ defined on $\mathbb{S}^n$, and $\mu$ the Lebesgue measure on $\mathbb{S}^n$.
How do I evaluate this? Do I just replace $d\mu (y)$ ...
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Is a set zero-measured when the lower dimensional truncated set of which is zero-measured?
I have no clue about this question, so here are some questions that may have answers or counter-examples. I hope this leads to more relevant results. If you know how to deal with any of them, or know ...
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39
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Bounded Convergence $\lambda$-a.e.
As the title says. So let $\{f_n\}_n$ be a sequence of Lebesgue measurable functions that converge $\lambda-$a.e. to a real valued function $f$ and let $E$ be Lebesgue measurable such that $\lambda(E)&...
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Characterization of measurable G such that $G \circ Z$ has a density with respect to Lebesgue Measure
Assume Z is a real valued random variable such that it's distribution has a density with respect to Lebesgue Measure (i.e. its Radon-Nikodym derivative w.r.t Lebesgue measure exists.) Then what is the ...
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1
answer
38
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Properties of Lebesgue points of a function
Let $F:\mathbb{R}^2 \rightarrow \mathbb{R}$ be such that for every $a,b \in \mathbb{R}$ the fuctions $F(a,\cdot), F(\cdot,b) \in L^1(\mathbb{R})$.
Consider the set $A:= \left\{a:\lim\limits_{r \...
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Integral over random Subset of $\mathbb{R}^2$? [closed]
What is the integral of a random subset $A$: $\int_A f \, d\lambda^2$.
I know what it would be if $A$ can be spil up into intervals but what if $A$ is not of some form of intervals?
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If $m(A)=0$, does this imply that $m(\log(A)) = 0$ when $A\subset \mathbb{R}_{+}$
The question is as follows:
Let $A\subset \mathbb{R}_{+}$ and let $\log(A)=\{\log(t): t\in A\}$.If $m(A)=0$,then is it true that $m(\log(A)) = 0$? If $m(A) < \infty$, then is it true that $m(A) <...
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1
answer
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Looking for a non-$\sigma$-finite measure for a counter example
This is just the context:
It can be shown, for a $\sigma$-finite measure $\mu$ on a measurable spaces $X$, the Lebesgue measure $\lambda$ on $[0,+\infty)$, and a measurable function $f:X\to [0,+\infty)...
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Let $f:[0,1]\mapsto [0,\infty]$ be a bounded, measurable function, prove that $\int_{[0,1]} f\text { }dm=\inf\int_{[0,1]} \phi\text { }dm$
Today, in class we were presented this theorem
Let $f:[0,1]\mapsto [0,\infty]$ be a bounded, measurable function,
prove that $\int_{[0,1]} f\text { }dm=\inf\{\int_{[0,1]}\phi\text{ }
> dm\text{ } |...
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1
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53
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A question related to a signed measure on the real numbers [closed]
I am currently working on the following problem from a past qualifying exam.
Let $\nu$ be a finite signed measure on $\mathbb{R}.$ Define a function $$g(x)=\int_{-\infty}^{x}\:d\nu:=\int \chi_{(-\...
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Subset $V$ Lebesgue measurable, but not Borel measurable
Define the function $g: [0, 1] \rightarrow [0, 1]$ with $$g(y) := \inf\{x \in [0, 1] : f(x) = y\},$$ where $f$ is the Cantor function.
Now let $V \subset [0, 1]$ be a set that is not Lebesgue ...
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Reference request: almost all level sets of $u : \Omega \subset \mathbb{R}^n, u \in L^1(\Omega)$ have measure zero
The following property appears in this other post:
If $\Omega \subset \mathbb{R}^n $ is a bounded set and $u \in L^1(\Omega)$, then $\mu(\{ x \in \Omega \mid u(x) = t\}) = 0$ for almost all $t \in \...
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Showing equality of first norms
We consider a function $f \in L_1[0,1]$, and define $g : [0,\infty) \to [0,1]$ in terms of $f$, where $g(y) = \lambda(|f|^{-1}(y,\infty])$ for $\lambda$ the Lebesgue measure, and $|f|^{-1}(y,\infty] = ...
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There is a function $f$ such that $f \in L^{p}$ and $f \notin L^{q}$ for $p,q \geq 1$ and $p\neq q$
Problem: For $\displaystyle{p \geqslant 1}$,
consider
\begin{align*}
f ( x ) = \frac{1}{ \displaystyle{ x^{ \frac{1}{p} } \big( \log^{2}{x} + 1 \big) } }
\qquad
\text{for all } x \in ( 0 , ...
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Show that the image of a homeomorphism involving the Cantor function has a certain measure
Currently working on the task of showing the following:
Let $f$ be the Cantor function and $$g: [0,1] \rightarrow [0,1], x \mapsto \frac{f(x)+x}{2}$$
Then $\lambda(g(C)) = \frac{1}{2}$
where $\lambda$ ...
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$\int_0^1 x^n f(x)\,\mathrm{d}x = 0$ for all $n$ implies $f=0$ almost everywhere
It has been shown that given $f \in \mathcal C[0,1]$, we have that if $\int_0^1 x^nf(x)\,dx = 0$ for all $n \in \mathbb N$, then $f = 0$.
I was thinking of generalizing this statement to $L^p$ spaces. ...
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Verifying the countably additive measure property of $\nu_f$ when $\nu_f(E)$ is infinite
I am studying measure theory and have come across a problem where I am asked to show that a certain function $\nu_f$ is a measure. Specifically, let $f \in L^+((X, \mathcal{M}, \mu))$; for any $E \in \...
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Nearly covering the complement of a nowhere dense set
Let $A\subseteq [0,1]$ be a nowhere dense subset of $[0,1]$ with Lebesgue measure $0$, and let $A'$ be its complement. I want to prove that, for any fixed small $\varepsilon>0$, we can find a ...
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Question About measure theory proof about subadditivity
My instructor showed us the following theorem and proof of it on class, but I have some question about circled part. 1st, why is there $\epsilon * 2^{-n}$ on the proof, secondly how could we replace $...
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Additivity of outer measure when one set is closed (Axler: Measure, Integration & Real Analysis)
Suppose $A$ and $F$ are disjoint subsets of $\mathbb{R}$ and $F$ is a closed set. Then $|A \cup F| = |A| + |F|$, where $|X|$ denotes the outer measure of a set $X$. In his proof, Axler (2.63) shows ...
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Is it possible that a set be measurable with a measure function but not with the other?
Consider we have a measurable subset $A$ of euclidean space $\mathbb{R}^p$ with respect to Lebesgue measure. Is it always measurable in respect to any measure function (I mean non-negative,finite,...
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1
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If $\int_a^b f(x) \ dx > 0$ then $f(x) > 0$ a.e?
Let $f$ be a measurable nonnegative function with domain $\mathbb{R}$, such that
\begin{equation}
\int_a^b f(x) \ dx > 0
\end{equation}
when $a < b$. Can we say that $f>0$ almost everywhere? (...
- 151
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0
answers
54
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Continuous Almost Everywhere Implies Measurability
Let $f \colon \mathbb{R} \to \mathbb{R}$ be continuous almost everywhere, i.e. the set of all points where $f$ is not continuous has Lebesgue measure $0$ (1 dimensional Lebesgue measure). Call this ...
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Can every subset of $\mathbb{R}$ a measurable set (with respect to some measure $\mu$)?
A problem occur to me when I was learning real analysis:
Can we construct a measure $\mu$ on $\mathbb{R}$ such that for all $E\subset\mathbb{R}$, $E$ is $\mu$-measurable?
I know that for Lebesgue ...
- 478
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0
answers
23
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equivalence of the $L^1$ norm and the TV norm in the space of integrable functions
Let $I:= [0,1]$ and let $L^1(I)$ be the space of integrable functions $f \colon I \to \mathbb{R}$, i.e. $f \in L^1(I)$ if and only if $\int_I|f| dm< \infty$.
For $f \in L^1(I)$, consider the two ...
1
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0
answers
54
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Lebesgue points of a.e. equal function
Let $f=g$ pointwise a.e. in $\mathbb{R}$.
Then what can we say about the Lebesgue points of $f$ and $g$?
Clearly the set of Lebesgue points of $f$ (denoted by $L_f$)and that of $g$ (denoted by $L_g$)...
- 596
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1
answer
38
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Outer Measure Equality and Limits: Why equality remains under limits for for limit of set?
I am reading measure theory in Rudin Book and in one of the its proofs, it is derived from an equality which exists between four sequences that this equality should also exist on four Limits. I think ...
- 111
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0
answers
43
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Are orbits of a Hamiltonian flow always measurable with measure zero?
Let $H: \mathbb{R}^{2n} \rightarrow \mathbb{R}$ be smooth and let $\phi^t$ be the Hamiltonian flow of $H$. Consider the restriction of the flow to a regular energy hypersurface, $\Gamma$, with the ...
0
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0
answers
25
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Equivalence of Assertions for Functions of Bounded Variation
I am trying to prove that the following three assertions about a function $f$ on an interval $[a, b]$ are equivalent, but I am encountering some difficulties:
$f$ is the difference of two increasing ...
0
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1
answer
41
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Arbitrarily close approximations of supremum and infimum
As I am studying real analysis a certain pattern keeps reoccurring - an example of this is from Measure Theory (2nd edition) by Cohn, Donald L., namely the proof of Lemma 1.5.3. The proof uses the ...
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0
answers
12
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cadlag functions equal a.e. implies they are equal everywhere
Let $f,g:\mathbb R\to\mathbb R$ be cadlag (continuous from the right and limits existing from the left) functions that are equal almost everywhere with respect to Lebesgue measure. I.e. $f(t+)=f(t)$ ...
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84
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Is this a known conjecture or a problem?
Let $K$ be any measurable set. If $B(0,a)$ is the ball such that $|B(0,a)| = |K|$ and $C_K$ is such that
\begin{equation}
(*) \quad a^2|K| = C_K\int_{B(0,a)}|x|^2dx = \int_{K}|x|^2dx,
\end{equation}
...