Tagged Questions
0
votes
1answer
17 views
Approximation for $L_{\text{loc}}^{\infty}(U)$ is this proof correct?
Let $U\subset\mathbb{R}^n$ be open and bounded. I am trying to extend Evans' proof (in his PDE book) for approximating functions in $L_{\text{loc}}^{p}(U)$ for the case that $p=\infty$ using ...
0
votes
0answers
17 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 ...
0
votes
0answers
34 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 ...
2
votes
1answer
59 views
About measure theoretic interior and boundary
Reference:- Evans-Gariepy, Federer, other books and notes of geometric measure thoery.
I just want to clarify whether these definitions of measure theoretic interior
and boundary are correct. Given ...
7
votes
1answer
83 views
Regular open set whose boundary has nonzero volume.
I found this question quite interesting, but its answers were disappointingly non-geometric. I'd be interested to know whether there exists a geometric example.
To be precise about what I mean by a ...
1
vote
0answers
30 views
Show derivative of integral equals integral of partial derivative if M[0,1]-measurable
I am trying to determine a method of approaching the following:
Suppose that $f:[0,1] \times (0,1)$ $\rightarrow$ $\mathbb{R}$ is such that, for each $y \in (0,1)$, the function $f^{[y]}(x) = f(x,y)$ ...
2
votes
1answer
62 views
What is the difference between the spaces $L^1$($\mu$) and $L^1$(d$\mu$)? And is one a subset of the other?
What is the difference between the spaces $L^1$($\mu$) and $L^1$(d$\mu$) ? And is one a subset of the other?
$\mu$ is the Lebesgue measure.
1
vote
1answer
98 views
Determining the Lipschitz constant
Determine the corresponding Lipschitz constant of $f(t,y(t))=e^{(t-y)/2}$, where $D=\{(t,y) : 0\leq t \leq 1,-\infty<y<+\infty\}$.
0
votes
0answers
72 views
Show $f$ is in $L^1$ (d$\mu$) space and $_X\int f $ d$\mu=\lim_{n\to \infty}\int_X f_n d\mu$
Suppose $f$ is in $L^1$($\mu$). Prove that for each $\epsilon > 0$ , there exists a $\delta > 0$ so that the $\int |f|\mathrm d\mu$ < $\epsilon$ over the set $E$ whenever $\mu(E) < ...
5
votes
0answers
125 views
Operator completly continuous
For $\lambda>0$, let $v(t)=\lambda \phi(t)$. Consider the BVP
consisting of the equation $$u'''=\lambda[f(t,[u-v]^*+\gamma)+M(t)] ,t\in (0,1)$$
and (BC):$u(0)=u'(p)=\int_q^1 w(s)u''(s)=0 ...
2
votes
1answer
35 views
about well-defined integral kernel
Let $\phi:\mathbb{R}^n\times\mathbb{R}^n\to\mathbb{R}$ measurable function such that
$$ \int_{\mathbb{R}^n}|\phi(x,y)|\ dx \leq M\ , \quad \int_{\mathbb{R}^n}|\phi(x,y)|\ dy \leq M\,.$$
Let $f\in ...
2
votes
1answer
51 views
Interchange differential operator with Lebesgue integral.
Under what condition am I able to interchange a differential operator with an integral? More precisely, given a function $f:\Omega\times U\to\Bbb R$ from a measure space $(\Omega,\mathscr A,\mu)$ and ...
6
votes
1answer
91 views
Image of a set of zero measure has zero measure
I am studying for my final and got stuck on the following problem from the previous year. I put my attempt below.
Suppose that $I\subset \mathbb{R}$ is an open interval, $f:I\rightarrow \mathbb{R}$ ...
2
votes
1answer
52 views
Concerning the noncompactness of the map, $I: W^{1, 2}(\mathbb{R}^n)\rightarrow L^{2^{\ast}}(\mathbb{R}^n)$
Consider the identity map $I:W^{1,2}(\mathbb{R^n})\rightarrow L^{2^{\ast}}(\mathbb{R}^n)$ where $n\geq 3$. Suppose that this map is not compact that is given some bounded sequence of functions ...
0
votes
0answers
28 views
Gap distribution independence proof
I have a question bout the proof of the independence of gap RVs. Given the independent exponentially distributed random variables $\xi_1$, $\xi_2$ ~ $\text{Exp}(\lambda)$, and a corresponding order ...
0
votes
1answer
31 views
Polar form for $f\in L^2(\mathbb{R}^n;\mathbb{C})$
I have some doubts in measure theory. Suppose $f\in L^2(\mathbb{R}^n;\mathbb{C})$, then $f=f_1+if_2$, where $f_1,~f_2\in L^2(\mathbb{R}^n;\mathbb{R})$. Is it possible to write this function in a polar ...
2
votes
1answer
34 views
Abstract integral - Borel measures - $L^p$ spaces
Let $(X,\mu,M)$ be a finite measure space. Suppose $T\colon X \to X$ is measurable and $\mu(T^{-1}E) = 0$ whenever $E \in M$ and $\mu(E)=0$. Prove that these exists $h \in L^1(\mu)$ such that $h ...
1
vote
3answers
71 views
A condition that balls have finite measure
Let $(X,d)$ be a metric space and let $\mu$ be a positive measure on $X$. I want to require that $(X,d)$ and $\mu$ have either of the following properties:
$\forall y \in X$, $\forall r \geq 0$, ...
1
vote
1answer
62 views
Part of proof 11.10 in Rudin's Principles of Mathematical Analysis
There is a part of proof 11.10 that I don't get in Rudin's Principles of Mathematical Analysis (3rd edition).
The whole theorem is the following two statements:
$\mathcal{M}\left(\mu\right)$ is a ...
0
votes
0answers
54 views
Riesz Representation Theorem and Indicator Function
I've been dealing with the Riesz Representation Theorem for measures and it is obvious that having a measure $\mu$ I can get a continuous linear functional $\mu^*$ in $C(X)^*$ where $X$ is a compact ...
1
vote
1answer
34 views
Analysis - Fourier Transforms - show that convolution of characteristic functions is continuous
I would appreciate any instruction on the following exercise from real and complex analysis:
Suppose $A$ and $B$ are measurable subsets of $\Re^1$, having finite positive measure. Show that the ...
0
votes
1answer
68 views
Whether convergence in L2 norm implies convergence a.e.? [duplicate]
How to prove or disprove$$\lim_{n\to\infty}\|f_n-f\|=0\;\Rightarrow \;\lim_{n\to\infty}f_n(x)=f(x)\; a.e.?$$ Any hint is appreciated.
0
votes
0answers
49 views
Disproving Littlewood's first principle
Im trying to disprove the principle for $\epsilon = 0$ so need to get measure of $(E\setminus G) \cup (G\setminus E)$ equal to $0$. Any suggestions for the counterexample?
1
vote
1answer
41 views
Lebesgue integrable function and limit
Show that if $f$ is a Lebesgue integrable function on $A\subset\mathbb R$ and $$A_n=\{x\in A:|f(x)|\geq n\}$$ for $n\in\mathbb N$, then $\lim_{n\to\infty} n\cdot m(A_n)=0$.
My solution which is ...
1
vote
1answer
34 views
Why does the supremum over finite partitions not suffice in defining total variation of complex measure?
In Rudin's Real and Complex Analysis, Chapter 6, eqn. 3, the total variation of a complex measure is defined as a supremum over all possible partitions of a set.
Why do we need to consider all ...
2
votes
1answer
42 views
How to recover a measure from its Fourier transform?
Let $f$ be the complex function defined on $\mathbb{R}$ by
$$
f(t)=\frac{1-it}{1+it}.
$$
1) Does there exist a complex bounded measure $\mu \in M(\mathbb{R})$ such that $\hat{\mu}=f$ (where $\hat{}$ ...
1
vote
4answers
36 views
Measure nonzero implies dense on a rectangle
This would be a very handy lemma for me but I have been unable to prove it thus far.
If $S \in \mathbb{R}^n$ is bounded and is not of measure zero, then there exists a rectangle $R$ such that $S$ ...
1
vote
0answers
40 views
Lebesgue measure of two functions.
Assuming Lebesgue Measure
If $f$ and $g$ are measurable functions, show that the set ${\{x:f(x)\gt g(x)\}}$ is measurable.
My idea so far is that since $f$ is measurable, the set ${\{x:f(x)\gt c\}}$ ...
0
votes
1answer
33 views
An alternative definition for integral of a nonnegative measurable function in terms of infimum
How could I show "integral of a nonnegative measurable function f could be defined as the infimum of a set of integrals of simple functions g with f<=g for all g".
We could assume f is bounded by ...
2
votes
1answer
43 views
Equivalence of measures and $L^1$ functions
Suppose we have two probability measures $\mu$ and $\delta$ on $(X, \mathcal{B})$ such that $ \delta <<\mu << \delta $. How can I prove that $f \in L^1(X,\mathcal{B}, \mu)$ iff $f \in ...
2
votes
1answer
108 views
Riemann-Stieltjes integrability criterion
I am currently reading through chapter 11 of Rudin's Principles of Mathematical Analysis, and I'm trying to solve problem 7:
Find a necessary and sufficient condition that $f \in \mathfrak R(\alpha)$ ...
0
votes
0answers
31 views
Conditions under which a real function is measurable [duplicate]
Consider a function $f: \mathbb R \to \mathbb R$ such that $|f|$ is measurable and $f^{-1}(F)$ is measurable for every finite set $F \subset \mathbb R$. Under which conditions will $f$ be measurable? ...
1
vote
1answer
77 views
Rudin Real and complex analysis question[Differentiation]
At the beginning of the chapter on differentiation, the following theorem is stated without proof. Apparently it is so trivial that it does not require justification. I however don't find it so ...
2
votes
1answer
60 views
Small question about a lemma of measurability
Hi ;
I have this lemma , and i want to ask tow questions :
1) What is the diffrence between say that $\varphi$ is measurable and to say that $\varphi$ is $(\mathcal{T},\mathcal{B}(U))$measurable .
...
3
votes
2answers
113 views
Measuring Vitali sets
If $V$ is a vector space and $m$ is the counting measure, then $$\dim(V) = \inf \{m(U) : U \subset V, \text{ span}(U) = V\}.$$
Given a measure space $(V, \mathcal M, \mu)$ such that $V$ is a vector ...
4
votes
1answer
43 views
Is the Minkowski sum of two Lebesgue measurable set measurable?
Let $E,F \subseteq \mathbb R$ be Lebesgue measrable. Is the sum $E+F=\{x+y:x \in E,y \in F \}$ Lebesgue measurable as well?
-2
votes
1answer
133 views
Let $f$ be a function $f:\Bbb R\to \Bbb R$ and $f^{−1} (F)$ for every $F$ subset of $\Bbb R$ and finite $F$ be measurable which is correct?
Let $f$ be a function $f: \mathbb{R} \to \mathbb{R}$ . $|f|$ is measurable and $f^{−1} (F)$ for every $F\subset \mathbb{R}$ is measurable too (and $F$ is finite) which of the following is correct?
...
0
votes
0answers
28 views
Question about probability measures on the real line [closed]
http://www2.imperial.ac.uk/~boz/M34P6/P11_2.pdf
Dear comrades. I am struggling with Ex 1.4(i) on here. I think that I have proved that the measures $\mu$, $\nu$, $\lambda$ are all equivalent, in the ...
2
votes
2answers
50 views
A question about a stronger version of Hölder inequality
Let $\lambda$ be the ordinary Lebesgue measure onto $(0,1)$. Let $a>1$, $f,g \colon (0,1) \to (0,+\infty)$ two $\lambda$-measurable functions and $B \subset (0,1)$ a $\lambda$-measurable subset ...
0
votes
1answer
39 views
Measurability of multifunctions
If $(T,\mathcal{A})$ is a measurable space , $X$ a meatrizable separable space.
$F$ a multifunction from $T$ to compacte subsets of $X$.
We want to prove that $F$ is measurable if
$\forall U\in X$ ...
0
votes
0answers
23 views
Equivalent definition of Riemann Stieltjes Integral
Given $\epsilon>0$, there exists $\delta>0$ such that $|R_{\Gamma}-R_{\Gamma^{'}}$|$<\epsilon$, if $|R_{\Gamma}$|, $|R_{\Gamma^{'}}$|$<\delta$ implies $\int_{a}^{b}fd\phi$ exists.
I ...
2
votes
1answer
37 views
If $\mu$ is a complex measure, every set $E$ has $A \subset E$ so that $|\mu(A)| \ge \frac{1}{\pi}|\mu|(E).$
If $\mu$ is a complex measure on a $\sigma$-algebra $M$, show that every set $E \in M$ has a subset $A$ for which $$|\mu(A)| \ge \frac{1}{\pi}|\mu|(E).$$
The suggestion is as follows:
Put ...
2
votes
1answer
55 views
If $f_k \to 0$ a.e. and $\sum_n n 2^n \mu\{|f_k| \in (2^{n-1}, 2^n]\} \leq 1$ for all $k$, then $\int f_k \to 0$.
(Stanford Real Analysis Qualifying Exam: Spring 2012) (Ideal time: 18 minutes)
(a) Let $\mu$ denote Lebesgue measure on $[0,1]$. Let $f_k\colon [0,1] \to \mathbb{R}$ be Lebesgue measurable ...
1
vote
0answers
108 views
Question on measurability
I need help to prove that :
If $(T,\mathcal{A})$ is measurable space, and $U$ a metric space , we say that $f:T\rightarrow U$ is (strongly) measurable if one of the following equivalent properties ...
3
votes
1answer
36 views
Inequality between product measure and its projection
$\newcommand{\smin}{\setminus}
\newcommand{\sset}{\subseteq}$If $\mu$ is a measure on $X$, $ \nu $ a measure on $Y, \gamma $ a measure on $X \times Y$ s.t. $ \gamma(A \times Y) = \mu (A) $ and ...
1
vote
1answer
29 views
Equivalent measures if integral of $C_b$ functions is equal
Is it true that if $X$ is a measure space and $\mu, \nu$ are Borel probability measures on $X$ if
$$ \int_X \phi \ d \mu = \int_X \phi \ d \nu \qquad \forall \phi \in C_b(X) \text{ (continuous and ...
1
vote
1answer
62 views
Question about integration (related to uniform integrability)
Consider a probability space $( \Omega, \Sigma, \mu) $ (we could also consider a general measure space). Suppose $f: \Omega -> \mathbb{R}$ is integrable. Does this mean that
$ \int |f| \chi(|f| ...
1
vote
2answers
62 views
Pushforward measure integral property
Let $\mu$ be a Borel measure on $X$ and $T: X \rightarrow Y$ a Borel map ($Y$ topological space). I define $$ T _* \mu(E) = \mu(T^{-1}(E)) \qquad \forall E \subset Y \quad \text{Borel} $$
It is easy ...
1
vote
1answer
111 views
$f = g$ a.e., $f$ is measurable, $g$ is not
Suppose ($X,\cal M, \mu$) is not complete. Let $E$ be a subset of a set of
measure zero that does not belong to $\cal M$. Let $f=0$ on $X$ and $g=\chi
_E$. Show that $f=g$ a.e. on $X$ while $f$ ...
1
vote
1answer
33 views
Quotient of two $L^1$ functions
Suppose that we have two $L^1(\mathbb{R})$ functions, $f$ and $g$, where $g$ is non-zero almost everywhere. Can we conclude that their quotient $\frac{f}{g}$ is finite almost everywhere?
