1
vote
1answer
17 views

Estimating the $(N-1)$- Hausdorff measure of $\Omega\cap \partial B(0,r)$ when $\lim_{r\to\infty} m(\Omega\cap B(0,r))/m(B(0,r))=0$.

Let $\Omega\subset\mathbb{R}^N$ be a open, unbounded and connected set ($N\ge 2$). Let $m$ and $\mathcal{H}^{N-1}$ denote respectively, Lebesgue and $(N-1)$-Hausdorff measures. Suppose that ...
0
votes
2answers
37 views

Why we can not extend Lebesgue outer measure for all subsets of real line?

I wonder that why we can't extend the Lebesgue outer measure for all the subsets of real line? Why we can't define another such measure on subsets of reals? I can't image why this happen. Counter ...
0
votes
1answer
37 views

A Set That Is “Precisely” Measure-Dense [duplicate]

This question asks for a set of real numbers that is measure-dense, whose complement is also measure dense. In terms of $[0,1]$, the question asks for an $S$ such that for every open interval $I$ we ...
1
vote
0answers
19 views

Non-section representation of an intersection of sets

Let $X,\bar X,Y$ be arbitrary sets and $A\subseteq X\times Y$, $\bar A\subseteq \bar X\times Y$ be arbitrary as well. Denote: $$ A_x :=\{y\in Y:(x,y)\in A\} $$ and similarly for $\bar A$. Consider a ...
0
votes
0answers
35 views

Prove that the function $g(·)$ is twice continuously differentiable and that $g′′(α) ≥ 0$ for all $α ∈ \mathbb{R}$, i.e.

Let $f$ be a real Lebesgue measurable function on the interval $[0, 1]$ such that $∥f∥∞ < ∞.$ For $α ∈ \mathbb{R}$ define a function $g(α)$ by $g(α) = \log \int_0^1\exp[αf(x)] dx .$ (a) Prove that ...
2
votes
1answer
53 views

Showing that a set is not infinite in measure

Suppose $f_n \geq 0$ for all $n \geq 1$, $f_n \to f$ a.e. on $[0, \infty)$ and there exists a constant $M>0$ such that $$ \sup\limits_{n} \int_{E} f_n(x)dx \leq M \mu(E)$$ for each measurable ...
0
votes
1answer
46 views

Find $\lim_{n \rightarrow \infty}$$\int_0^n(1 + \frac{-x}{n})^n\cos(\frac{x}{\sqrt{n}})e^{x/2}dx$

Find $\lim_{n \rightarrow \infty}$$\int_0^n(1 + \frac{-x}{n})^n\cos(\frac{x}{\sqrt{n}})e^{x/2}dx$ I want to use dominated convergence theorem obviously. However, not sure how to dominate it. ...
2
votes
1answer
47 views

Differentiation under the integral if and only if we have an $L^1$ dominator

Let $f(x)\in L^2(\mathbb{R})$ and define $$g(t) = \int_\mathbb{R} f^2(x)\exp(-tx^2)dx$$ for $t\geq0$. We want to show that $g(t)$ is continuously differentiable if and only if $xf(x)\in ...
0
votes
1answer
47 views

Relation between Sobolev Space $W^{1,\infty}$ and the Lipschitz class

I have a Sobolev space related question. In the book 'Measure theory and fine properties of functions' by Lawrence Evans. I know the result that states that for $f: \Omega \rightarrow \mathbb{R}$. $f$ ...
1
vote
1answer
31 views

The functions $\{f_n(x) = n\}$ are analytic and each miss the points $-2, -3$. But, they are not a normal family. So what am I missing. Thanks.

Here is a theorem of Montel: Let $\mathcal{F}$ be a family of analytic functions defined on a domain $\Omega$ . If there are two fixed complex numbers $a$ and $b$ that are omitted from the range of ...
1
vote
0answers
43 views

Prove that $f\ast g$ is defined a.e., integrable, and such that $∥f\ast g∥_1 ≤ ∥f∥_1 · ∥g∥_1$

Let $f,g : \mathbb{R} → \mathbb{R}$ be $L_1$-functions. Set $h(x) = \int_\mathbb{R}f (x − y)g(y) \, dm(y).$ Prove that $h(x)$ is defined a.e., $h ∈ L_1(\mathbb{R})$ and $∥h∥_1 ≤ ∥f∥_1 · ∥g∥_1.$ So I ...
1
vote
1answer
29 views

variation of a function over countable intervals

Let $f$ be a function of bounded variation on $[0,1]$. Let $\{[a_n,b_n]\}_{n=1}^\infty$ such that $(a_n,b_n)$ are pairwise disjoint and $\cup_{n=1}^\infty [a_n,b_n]=[0,1]$. (for example, $[1/2, 1], ...
3
votes
1answer
105 views

Product of Lebesgue and counting measures

Let $\mathbb R$ be endowed with the standard Euclidean topology and let $\widetilde {\mathbb R}$ denote the line endowed with the discrete topology. Let $\mu$ and $\nu$ denote the Lebesgue and ...
3
votes
0answers
30 views

Let $(X,μ)$ be a measure space. Find a necessary and sufficient condition on $(X,μ)$ that $L_q(E) ⊂ L_p(E)$ for all $1 ≤ p < q ≤ ∞.$

Let $(X,μ)$ be a measure space. Find a necessary and sufficient condition on $(X,μ)$ that $L_q(E) ⊂ L_p(E)$ for all $1 ≤ p < q ≤ ∞.$ I want to say that the condition is that $E$ is finite. This ...
2
votes
0answers
35 views

Show that $\int_E f (x, y) dx$ is differentiable with respect to $y$ and $\frac{d}{dy}\int_E f(x,y)dx=\int_E \frac{d}{dy}f(x,y)dx.$

Assume that $f = f(x,y)$ is a function defined on $E × (a,b).$ For each fixed $y ∈ (a,b),$ $f$ is integrable with respect to $x$ on $E$, and for each fixed $x ∈ E$, $f$ is differentiable with respect ...
1
vote
1answer
31 views

Proving and visualizing $\mathbf 1_{(x,x+a]}(y) = \mathbf 1_{[y-a,y)}(x)$

Here is a trick from one of the proofs in probability: $$\iint \mathbf 1_{(x,x+a]}(y) \ \lambda(dx) \ \mathbb P(dy) = \iint \mathbf 1_{[y-a,y)}(x) \ \lambda(dx) \ \mathbb P(dy)$$ for $a>0$. So ...
3
votes
0answers
57 views
+50

What types of integrals cannot be solved using improper Riemann-Stieltjes Integration?

I came across the wikipedia discussion of the Riemann-Stieltjes integral. The first sentence in the "Generalization" section gave me pause: An important generalization is the Lebesgue–Stieltjes ...
2
votes
1answer
32 views

Does the limit $\lim_{k\to\infty}\int|\cos kx |f(x) d\lambda(x)$ always exist?

Let $f$ be a Lebesgue integable function. Does the limit $$\lim_{k\to\infty}\int|\cos kx |f(x) d\lambda(x)$$ always exist?
4
votes
2answers
53 views

Borel measure supported on $\mathbb{Q}$

Let $\mu$ be a Borel measure supported on $\mathbb{Q} \subset \mathbb{R}$. Must $\mu$ be a sum of Dirac measures?
3
votes
0answers
39 views

Measurability of $\int f(x,\bullet) d\nu$ (product space)

I'm self-studying a set of notes on measure theory. For this chapter, the author said that he is not going to prove every theorem, and I have some trouble with this particular one: Theorem 5.15. Let ...
0
votes
2answers
44 views

Representation of linear functionals on a certain Banach space

Let $C^k([0,1])$ be the space of such complex-valued functions on $[0,1]$ that are continuously differentiable at least $k$ times ($k\in\mathbb N$). It is well known that $C^k([0,1])$ is a Banach ...
3
votes
1answer
66 views

$f$ is in $L^p$ iff sum is finite

Let $p\in [1,\infty)$.Prove that $f\in L^p(\mu)$ if and only if $\sum_{n=1}^\infty(2^n)^p\mu (\{x:|f(x)|\gt2^n\})\lt \infty.$ My idea, I assume measure is finite, I wrote ...
5
votes
1answer
38 views

Let $f_n:\mathbb{R}\rightarrow [0, 1]$ be functions such that $\sup_{x \in \mathbb{R}}f_n(x) = 1/n$ and $||f||_1 = 1$.

Let $f_n:\mathbb{R}\rightarrow [0, 1]$ be functions such that $\sup_{x \in \mathbb{R}}f_n(x) = 1/n$ and $||f_n||_1 = 1$. Set $F(x) = \sup_{n \in \mathbb{N}}f_n(x)$. Prove that $\int_\mathbb{R}F(x)dx ...
3
votes
1answer
18 views

$L^p$ integral on every measurable subset of $\Bbb R$

Suppose $f:\Bbb R \to \Bbb R$ is in $L^p$ for some $p>1$ and also in $L^1$. Prove there exist constants $c>0$ an $\alpha \in (0,1)$ such that $\int_A|f(x)|dx\le cm(A)^{\alpha}$, for every ...
1
vote
1answer
29 views

Caratheodory extension theorem: which is the “unique extension”

According to Wikipedia, the constructed measure on $\sigma(R)$ is a unique extension. However, in most situations, the $\sigma$-algebra of Caratheodory-measurable sets $M$ is larger than $\sigma(R)$. ...
1
vote
1answer
23 views

On finite measurable space $X$, the whole of $L^p(X)$ is closed in $L^1(X)$ iff there is a const $C$ st $||f||_p\leq C||f||_1, \forall f \in L^p(X)$

On finite measurable space $(X, \mathcal{M}, \mu)$, the whole of $L^p(X, \mu)(p>1)$ is closed in $L^1(X,\mu)$ iff there is a const $C$ st $||f||_p\leq C||f||_1, \forall f\in L^p(X)$, iff both ...
3
votes
0answers
45 views

Image of Cantor set under Cantor-Lebesgue function

Let $m^{\ast}$ be the Lebesgue outer measure and $m$ the Lebesgue measure. Let $\phi$ be the Cantor Lebesgue function and let $\psi(x) := x + \phi(x)$. Let $C$ be the standard Cantor set, why does ...
1
vote
1answer
36 views

Approximation functions in $L^{1}$ by indicator functions of dyadic cubes

Let $\mu$ be a finite positive regular Borel measure on $\mathbb{R}^{d}$ and let $S$ be the family of finite unions of squares of the form $\{a_{1}2^{n} \leq x_{1} \leq (a_{1} + 1)2^{n}, \ldots, ...
4
votes
3answers
106 views

Find $\delta >0$ such that $\int_E |f| d\mu < \infty$ whenever $\mu(E)<\delta$

I am studying for a qualifying exam, and I am struggling with this problem since $f$ is not necessarily integrable. Let $(X,\Sigma, \mu)$ be a measure space and let $$\mathcal{L}(\mu) = \{ \text{ ...
0
votes
0answers
32 views

Singular measures

Suppose $\mu$ and $\nu$ are two signed measures and they are mutually singular. I heard 2 version on this definition there $\exists N $ s.t.$\mu(N)=0$ and (1) $\nu(N^c)=0$ (2)$|\nu|(N^c)=0$ which ...
1
vote
1answer
40 views

A problem in the proof of Jordan decomposition theorem

How to obtain the red rectangle in the picture? thanks in advance.
0
votes
1answer
37 views

Denseness and non-measureability

I made up a question for myself and tried to answer it. I'm not completely sure of my question and answer, since I lack a grounding in analysis (the tragedy of doing physics then mathematical ...
2
votes
1answer
61 views

Can we find uncountably many disjoint measurable subsets of $\mathbb{R}$ with strictly postive Lebesgue measure?

Can we find uncountably many disjoint measurable subsets of $\mathbb{R}$ with strictly positive Lebesgue measure?
0
votes
1answer
32 views

Converge in measure implies converge a.e if $f_n$ are monotone

Let $\{f_n\}$ be monotone sequence of functions such that $f_n$ converges in measure to $f$. Is it true that $f_n$ converges to $f$ a.e$?$ I am sure it has a sub-sequence that converges to $f$ a.e. ...
2
votes
2answers
90 views

Do we need the $f,g \geq 0$ condition for $\int f \ d\mu = \int g \ d\mu$?

My lecture notes state the following corollary: Let $f,g \in \mathcal M_\bar{{\mathbb R}}$ (that is, numerical measurable functions), $f=g$ $\mu$-almost everywhere and $f,g \geq 0$. Then $\int f \ ...
12
votes
2answers
171 views

Do differentiable functions preserve measure zero sets? Measurable sets?

Consider Lebesgue measure on $\mathbb{R}$ and let $f:\mathbb{R}\to\mathbb{R}$ be differentiable. Does $f$ necessarily preserve measure zero sets? Does $f$ necessarily preserve measurable sets? ...
5
votes
1answer
63 views

$\int_0^1f(x)dx = 2, \int_0^1g(x)dx = 1, \text{and} \int_0^1[f(x)]^2 dx ≤ C$ for some constant $C > 4.$

Suppose $f$ and $g$ are nonnegative measurable functions on the interval $[0,1],$ with the properties $$\int_0^1 f(x)\,dx = 2, \int_0^1g(x)\,dx = 1, \text{ and }\int_0^1[f(x)]^2 dx \le C$$ for some ...
1
vote
0answers
51 views

If $f_{n}\rightharpoonup \bar{f}$ and $f_{n}(x) \rightarrow f(x)$ pointwise a.e., then is $\bar{f} = f$ a.e.? [duplicate]

Suppose $f_{n}$ is a sequence of functions in $L^{p}(\mathbb{R}^{d})$ such that $\|f_{n}\|_{L^{p}} \leq 1$ for all $n$ and $f_{n}(x) \rightarrow f(x)$ pointwise almost everywhere as $n \rightarrow ...
0
votes
2answers
26 views

Show that $\int_X gdν=\int_X gfdμ$ for all $g∈L_1(ν).$

Let $μ$ and $ν$ be finite (positive) measures on a measurable space $(X, M),$ and suppose that $ν(E)=\int_E fdμ$, for all $E∈M,$ $E$ where $f$ is some function in $L_1(μ).$ Show that $\int_X ...
4
votes
1answer
33 views

Locally integrable function with a uniform bound…

I'm a bit lost... I have a measure space $(\Omega,\mathcal{B}(\Omega),\mu)$ where $\mathcal{B}(\Omega)$ is a Borel set. Let $f$ be a real-valued measurable function on $\Omega$ and $\mathcal{K}$ be ...
2
votes
1answer
34 views

Clarification about a basic proposition about measurable functions

I am making my way through "Linear Functional Analysis" by Bryan P.Rynne and Martin A.Youngson (second edition). Given a measure space $(X,\Sigma ,\mu )$ we define a function $f$ to be measurable if ...
0
votes
1answer
55 views

Measure extension theorem(unique) [closed]

Please give an example of two probability measures $\mu \not = \nu$ on $\cal{F} $= all subsets of {1, 2, 3, 4} that agree on a collection of sets C with $\sigma(C)=\cal{F}$ . thanks in advance.
0
votes
1answer
40 views

Is the diagonal measurable

Suppose we have $X=Y=[0,1]$, $\lambda$ the Lebesgue measure on $[0,1]$, and $\nu$ the counting measure on $[0,1]$. Show that the diagonal $\Delta=\{(x,x):x\in X\}$ is $\lambda\times\nu$-measurable. I ...
1
vote
0answers
29 views

boundedness of $\{\int_E(g f_n)\}$ implies boundedness of $\{f_n\}$

I need some help on this problem: Let $E$ be a measurable set, $1 \le p < \infty$ and $q$ is the conjugate of $p$. Suppose that $\{f_n\}$ is a sequence in $L^p(E)$ such that for each $g \in ...
7
votes
0answers
79 views

Exercise on Radon measures, constructing a convergent sequence

Let $\mu$ be a Radon measure on $\mathbb{R}^n$ such that $\mu(B(0, s)) > 0$ for all $s > 0$ and suppose that $$C = \limsup_{s\ \downarrow\ 0}\frac{\mu(B(0, 2s))}{\mu(B(0, s))} < ...
0
votes
1answer
37 views

When does the convergence of the regularization of a function is decreasing?

Hi everyone: Let $\theta(x)$ equal $k\exp\left(-\frac{1}{1-\|x\|^2} \right)$ if $\|x\|<1$, and equal $0$ if $\|x\|\geq1.$ Here $\|\cdot\|$ designates the Euclidean norm in $\mathbb{R}^n$, and the ...
1
vote
0answers
51 views

Different definitions of uniform integrability

In different books and resources, I saw different definitions of uniformly integrable. For example, in some books the definition is like: Definition 1: $\{f_k\}\in L^1(E)$ is uniformly integrable ...
1
vote
1answer
44 views

Union of Increasing Sequences of Monotone Classes is not a Monotone Class.

In my text, we define a Monotone Class $\mathcal{M}$ of a non-empty set $X$ to be a collection of subsets of $X$ that is closed under monotone limits: that is, $(1)$ if $A_{i} \uparrow A$ with $A_{i} ...
2
votes
2answers
48 views

Can any measure be made into a bounded measure?

Is it possible to derive a bounded measure from any measure on a measure space? For example can the Lebesgue measure be made into a probability measure?
1
vote
2answers
30 views

Approximation in non-compact interval

Suppose that $f$ is a continuous function defined on a interval $I\subseteq \Bbb R$. (a) If $I=[0,1]$ and $\epsilon \gt0$ is given show that there are finitely many constants $a_k$, $1\le k \le n$, ...