Questions relating to measures, measure spaces, Lebesgue integration and the like.
1
vote
1answer
58 views
Show that a function is continuous
Let K be bounded and continuous and bounded on $\mathbb{R}^{n}$ and let $f$ be Lebesgue integrable on $\mathbb{R}^{n}$.
Show that the function $g$ defined on $\mathbb{R}$ by
$g(t) = ...
0
votes
1answer
36 views
Can't establish a lower bound on a supremum
I have a sequence of functions $f_{k,j}:[0,1]\to\mathbb{R}$ defined by
$$f_{k,j} = k^{\frac{1}{p}}\chi_{(\frac{j-1}{k},\frac{j}{k})},$$
for all $k\geq 1,1\leq j\leq k$.
This serves as an example of ...
0
votes
1answer
36 views
Measuring distances on any coordinate system
I was reading the book The ABC of Relativity from Betrand Russell, and at some point, the author mentions a method for measuring the distance between 2 points on any coordinate system. He says that ...
0
votes
2answers
46 views
Riemann integral and Lebesgue integral
$f:R\rightarrow [0,\infty)$ is a Lebesgue-integrable function. Show that
$$
\int_R f \ d m=\int_0^\infty m(\{f\geq t\})\ dt
$$
where $m$ is Lebesgue measure.
I know the question may be a little dump.
...
-1
votes
1answer
84 views
$f:[a,b]\to\mathbb{R}$ be a measurable function Then which of the following are true? [closed]
$f:[a,b]\to\mathbb{R}$ be a measurable function Then which of the following are true?
1.$\int_{c}^{d} f(x) dx=0\forall a\le c<d\le b$ Then $f=0$ a.e
2.$\int_{a}^{c} f(x) dx=0\forall a\le c\le b$ ...
2
votes
0answers
36 views
Egorov's Theorem for $\mathbb{R}^n$
Let $g,f,f_n:\mathbb{R}^n\to \mathbb{R}$ and $\{f_n\}_{n=1}^\infty$ be a sequence of functions that have the following property:
i) $f_n \to f$ a.e
ii)$|f_n(x)|\leq g(x)$ a.e and $g \in ...
2
votes
1answer
49 views
Weak $L^{p}$ spaces are quasi-normed?
Let $(X,\mathcal{B}, \mu)$ be a measure space. Then for $0< p < \infty$ by definition
$L^{p,\infty}(X,\mathcal{B}, \mu)$ is the class of all measureable functions $f$ such that
...
3
votes
2answers
71 views
Proof of Egoroff's Theorem
Let $\{f_n \}$ be a sequence of measurable functions, $f_n \to f$ $\mu$-a.e. on a measurable set $E$, $\mu(E) < \infty$. Let $\epsilon>0$ be given. Then $\forall \space n \in \mathbb{N} \space ...
1
vote
1answer
61 views
Why is ess sup $f$ not ess max $f$?
Consider a measure space $(X,\Sigma\,\mu)$. Given that one can easily prove that, $\mu$-a.e., $f \leq \text{ess} \sup_X f$, why is the notation not simply "$ \text{ess} \max_X f$"?
(Here $\text{ess} ...
0
votes
1answer
68 views
How many Borel conjectures are there
The following may be referred to as Borel conjecture:
Every strong measure zero set of reals is countable.
On the other hand Wikipedia refers to the following as the Borel conjecture:
Let $M$ and ...
1
vote
2answers
43 views
Definition of Lebesgue-Stieltjes measure on $\mathbb R$
Let $F:\mathbb R\to\mathbb R$ be a non-decreasing, left-continuous function. Let $a,b\in\mathbb R$, then define the Lebesgue-Stieltjes measure
$$ m[a,b]=F(b+)-F(a), \quad m(a,b)=F(b)-F(a+) $$
...
1
vote
1answer
67 views
Proving that for $f\geq0$ on $X$, $\int_X f d\mu = 0$ iff $f = 0$ a.e.
Okay, so the question is the following:
Suppose $f \geq 0$ is a measurable function on the measure space $(X,\Sigma,\mu)$. Prove that
\begin{align} \int_X f d\mu = 0 \text{ if and only if } f = 0 ...
1
vote
0answers
38 views
lebesgue measure of difference of two sets
Suppose $A,B$ are in $\mathbb R^n$, $A$ lebesgue measurable and $|(A \setminus B)\cup(B\setminus A)| = 0$. Show $B$ must be lebesgue measurable as well and $|A| = |B|$ (where $|\cdot |$ is the ...
1
vote
1answer
24 views
measure of limsup positive for infinitely many terms
let $(X, M, \mu)$ be a measure space, and let ${E_k}$ be in $M$. if $\mu(\cup E_k) < \infty$, and $\mu(E_k) \ge c > 0$ for infinitely many $k$'s for some $c$, then $\mu(\limsup E_k) > 0$. ...
1
vote
1answer
32 views
$g(x) = 0$ if $f(x)$ is rational measurability
Say $f$ is a measurable real-valued function on $X$, and put $g(x) = 0$ if $f(x)$ is rational and $g(x)=1$ if $f(x)$ is irrational. Is $g$ measurable?
1
vote
1answer
50 views
Composition of Lebesgue measurable function $f$, with a continuous function $g$ having a certain property, is Lebesgue measurable
Suppose that $f$ is Lebesgue measurable and $g$ is real valued, continuous, and has the property that for any null set $N$, $g^{-1} (N)$ is measurable. Then $f \circ g$ is also Lebesgue measurable. ...
6
votes
1answer
240 views
Need an explanation of this paragraph “Measure Theory”
I will just quote a part of one proof in "On uniformly regular topological measure spaces by Babiker: page 781" vol43 No4 Duke Math. J. 1976.
Let $I$ be the unit interval endowed with Lebesgue ...
2
votes
1answer
26 views
Uniformly regular measure “Babiker”
A regular Borel (Radon) probability measures $\mu$ on compact Hausdorff space $X$ is called uniformly regular if:
There is a countable family $\mathcal{A}$ of compact $G_\delta$-subsets of $X$ ...
1
vote
1answer
24 views
Is this function measurable? Something to do with Bochner space and norms.
Suppose $f:[0,T]\to X$ is a measurable map where $X$ is Hilbert space. Suppose also that $R(t):X \to X^*$ is an isometric isomorphism with
$$\lVert R(t)f(t)\rVert_{X^*} = \lVert f(t) \rVert_X$$
also ...
1
vote
1answer
54 views
A question about “nice” functions
Let $f:\mathbb R \rightarrow \mathbb R$ be a function such that $\lambda(I)=\lambda(f(I))$ for each interval $I \subseteq \mathbb R$. ($\lambda$ is Lebesgue measure here.) Let us call such functions ...
1
vote
0answers
26 views
Area of a Random Polygon
The following is a long description of a computation I'd like to make. You can think of the process described as a spider randomly building a web. I'd like to know how big we can expect the web to ...
1
vote
1answer
55 views
Integral over null set is zero but integral of Dirac delta function is 1
We know integral of any function over a null set is zero.
But for Dirac delta function ($\delta=+\infty$ iff $x=0$ otherwise $\delta=0$)
$$
\int_{-\infty}^{+\infty}\delta =\int_0^0\delta =1.
$$
Is it ...
1
vote
2answers
65 views
If $\lVert f(t) \rVert:[0,T] \to \mathbb{R}$ is measurable, is $f$ measurable?
Let $f:[0,T] \to X$ be a mapping to a Banach space $X$. If its norm $\lVert f(t) \rVert$ is measurable, is $f$ itself measurable? The converse is true.
1
vote
1answer
53 views
Exercise on Lebesgue measure ( Treatise of Analysis Vol2 by Dieudonné)
Someone challenge me to bring the solution from anywhere! So I have posted here and see, I am optimist because this website is excellent and its members are so helpful.
Let me start with this ...
2
votes
1answer
37 views
Gauss–Ostrogradsky formula for Distributions
Let $\Omega\subset\mathbb{R}^N$ be a bounded domain and $u\in W^{1,p}(\Omega)$, $v\in L^{p'}(\Omega)^N$ with $p\in (1,\infty)$. Let $\operatorname{div}(v)$ be the divergence of $v$ in the sense of ...
4
votes
3answers
85 views
A basic doubt on Lebesgue integration
Can anyone tell me at a high level (I am not aware of measure theory much) about Lebesgue integration and why measure is needed in case of Lebesgue integration? How the measure is used to calculate ...
1
vote
1answer
34 views
Measure of the set of real numbers that can be approximated in this way
Let $$A = \{x \in \mathbb{R}\mid \exists\,\text{infinitely many pairs of integers $p,q$ such that $|x-p/q| \leq 1/q^3$}\}.$$ Is the measure of $A$ equal to $0$? Any ideas?
2
votes
2answers
57 views
$P[X=Y]=0$ if $X,Y$ are i.i.d. with continuous c.d.f.
I am having lots of trouble proving the following statement:
Let $X,Y$ be two real valued random variables on a probability space $(\Omega,\mathcal{F},P)$. These two variables are independent and ...
3
votes
2answers
72 views
Baire's theorem from a point of view of measure theory
According to Baire's theorem, for each countable collection of open dense subsets of $[0,1]$, their intersection $A$ is dense. Are we able to say something about the Lebegue's measure of $A$? Must it ...
0
votes
0answers
22 views
Question on a third-order boundary value problems
This is the corollary $2.1$, from the article "Positive solutions of third order semipositone boundary value problems"
if $$u'''=\lambda \left(\sum_{i=1}^m c_i(t)u^{\mu_i}-d(t)\right)+e(t), t\in ...
1
vote
0answers
62 views
For what $p$ is $x^p$ Lebesgue Integrable?
Revising for an exam on Monday any help with the following question would be greatly appreciated;
If $f$ is a function on $(0, \infty)$ taking values in $\mathbb R$, defined $f(x)=x^p$ ($p$ is a real ...
1
vote
1answer
27 views
$\lim_{y \to \infty}\int_{R}f(x-t)\frac{t}{t^2 +y^2}dt=0?$ for $f\in L^{p}$, $p \in [1,\infty)$
For $f\in L^{p}$, $p \in [1,\infty)$
we want to prove:
$$\lim_{y \to \infty}\int_{R}f(x-t)\frac{t}{t^2 +y^2}dt=0$$
I'm not sure whether we can exchange the limit and the integral, cuz I cannot find ...
0
votes
0answers
33 views
Measurability of multifunction
Let $f:[a,b]\times \mathbb{R}^{n} \times \mathbb{R}^{m} \rightarrow \mathbb{R}^{n}$. Suppose $ f (.,x, u) $ is Lebesgue measurable for each $(x,u)$. Suppose also that $ f $ is continuous at $ (x, u) $ ...
16
votes
5answers
274 views
Examples of properties that hold almost everywhere, but that explicit examples unknown.
In measure theory one makes rigorous the concept of something holding "almost everywhere" or "almost surely", meaning the set on which the property fails has measure zero.
I think it is very ...
0
votes
2answers
30 views
Two random variable with the same variance and mean
Let $Y\in L^{2}(\Omega,\Sigma,P)$ and let $E[Y^2|X]=X^2$ and $E[Y|X]=X$. Could we prove that $Y=X$ almost surely.
My partial answer:
By the definition of conditional expectation we have ...
3
votes
0answers
51 views
Alternative rigorous definition of a surface integral
Consider some open subset $U$ of $\mathbb{R}^n$ where $U$ has a (piecewise) $C^1$-boundary. Let $f$ be some smooth (enough) real function. Is there some way to give a measure-theoretic definition of ...
4
votes
2answers
83 views
Optimal probability measure
Let $A$ be a finite set and let $\Bbb P$ be a probability measure on $A^{\Bbb N_0}$. Further, let $x_i:A^{\Bbb N_0}\to A$ be projection maps, so that $(x_i)_{i=0}^\infty$ can be treated as a ...
3
votes
1answer
62 views
Measures on all subsets of $\aleph_0$
A theorem of Ulam says:
A finite measure $\mu$ defined on all subsets of a set of cardinality $\aleph_1$ must be $0$ for all subsets if it sends every $1$-element subset to $0$.
Will this ...
1
vote
0answers
25 views
Lebesgue measure of set $M = \{ [x,y] \in \mathbb{R}^2; 2 < x + y < 3; x < y < 3x \}$?
although we can do this by splitting the area four ways and computing four integrals, my book suggests that I try the substitution $ u = x + y$ and $ v = \frac{y}{x}$.
I expressed $x$ and $y$ in ...
0
votes
0answers
36 views
Proving that $\bigotimes_{i=1}^n \cal{B}_{X_i} = \cal{B}_{X}$
Theorem: Given separable metric spaces $X_1,\ldots,X_n$ and $X=\prod_{i=1}^n X_i$, where $X$ has the product metric $d(f,g)=\sqrt{d_1 (f(1),g(1))^2 +\cdots + d_n (f(n),g(n))^2}$. Then ...
1
vote
1answer
29 views
Inequality between 2p-norm and p-norm for random variables
Recently I was studying something about random matrix theory, and class of sub-guassian / sub-exponential random variables is of interest. In the literature it gave an inequality as following:
...
3
votes
1answer
45 views
The semifinite portion of a measure $\mu$
Let $\mu$ be a measure and define $\mu_1$ such that $\mu(E)=\mu_1(E)$ for $\mu(E)$ finite. And for $\mu(E)$ infinite definite $\mu_1$ such that:
(i) if $E$ contains finite subsets of arbitrarily ...
2
votes
4answers
103 views
Book Suggestions for an Introduction to Measure Theory [duplicate]
Couldn't find this question asked anywhere on the site, so here it is! Do you guys have any recommendations for someone being introduced to measure theory and lebesgue integrals?
A mentor has ...
0
votes
1answer
31 views
A Measure For The Space of Probability Density Functions
Consider the space of all joint probability density functions of two variables. I want to know what the measure is of the portion of this space that is filled by uncorrelated joint pdfs relative to ...
4
votes
1answer
53 views
Measurability of an Indexed Product-Measure
If for any fixed $\omega_1$, $P_{\omega_1}$ is a probability measure and $Q_{\omega_1}$ is a stochastic kernel and both are measurable in $\omega_1$, is the indexed product measure ...
5
votes
3answers
162 views
why measure theory
I studied elementary probability theory. For that, density functions were enough. What is a practical necessity to develop measure theory? What is a problem that cannot be solved using elementary ...
1
vote
1answer
33 views
Does $u\in L^p(B)$ implies $u_{|\partial B_t}\in L^p(\partial B_t)$ for almost $t\in (0,1]$?
Let $B$ be the unit ball in $\mathbb{R}^N$ with center in origin and consider the space $L^p(B)$ with Lebesgue measure ($1<p<\infty$). Let $B_t\subset B$ be a concentric ball of radius $t\in ...
1
vote
1answer
60 views
Isomorphism Subalgebra
Given, the unit interval $I$ endowed with the Lebesgue measure $\mu$, and let $A$ be the (Boolean) algebra of Jordan measurable subsets of $X$ with respect to $\mu$, (i.e. those sets that satisfying ...
2
votes
1answer
54 views
Simplification of an expression
How do I simplify the following expression?
$$\displaystyle \frac{\int_q^1 w(s) \int_0^s e(\xi) d\xi ds}{2\int_q^1 w(s) ds} p$$
where $w(t)$ is nondecreasing $w(t)>0$ on $(q,1]$ , $e ...
3
votes
1answer
34 views
Are the continuous functions on $G$ dense in $L^{1}(G)$?
If $G$ is a locally compact group, is the set $C_{c}(G)$ of all continuous functions on $G$ with compact support dense in $L^{1}(G)$?
