Questions related to measures, sigma-algebras, measure spaces, Lebesgue integration and the like.

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2
votes
1answer
57 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 ...
1
vote
1answer
68 views

$F(x)=\int^{x}_{a} f(y) dy$ continuous (Lebesgue Integral)

Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be Lebesgue integrable and let $a \in \mathbb{R}$. We wish to show that $F(x)=\int^{x}_{a} f(y) dy$ continuous. I know, of course, what we need. We need to ...
0
votes
1answer
48 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
43 views

Need help with application of Hardy-Littlewood inequality (Marcinkiewicz space and distribution functions)

I am going over this work here. I couldn't understand the equality where the Hardy-Littlewood inequality is used. I think $\delta$ here is a weight so we can take it to be $1$ for simplicity. Would ...
2
votes
1answer
65 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 ...
1
vote
1answer
61 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
11 views

Is $\text{Id} = \chi_{\{ |u| \leq k\}} + \chi_{\{|u| > k\}}$ well defined for $u \in L^p(0,T;L^q)$?

Is the decomposition $$\text{Id}(z) = \chi_{\{ |u| \leq k\}}(z) + \chi_{\{|u| > k\}}(z)\tag{1}$$ well defined for $u \in L^p(0,T;L^q(\Omega))$? I guess (1) holds a.e. So the problem is, is the set ...
0
votes
0answers
31 views

$\int_{A}{f_{n}} \> d\mu \rightarrow \int_{A}{f} \> d\mu$ for each $A \in \mathfrak{M}$ given certain conditions [duplicate]

Let $(X, \mathfrak{M}, \mu)$ be a measure space. Assume the following items: $f_n$ is non-negative and integrable for each $n$, $f_n \rightarrow f$ almost everywhere, $\int{f_{n}} d\mu \rightarrow ...
1
vote
1answer
32 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 ...
5
votes
1answer
72 views

Measures which cannot be uniquely written as the sum of a purely atomic measure and a nonatomic measure

Maharam's theorem says that every complete measure can be written as the sum of a purely atomic measure and a nonatomic measure. According to the paper "Atomic and Nonatomic Measures" by R.A. Johnson, ...
3
votes
2answers
69 views

Without Lebesgue

Everyone knows following problem. Let $f$ be positive function on $[0,1]$ and there exist $I = \int_{0}^{1}f(x)dx$. Prove that $I>0$. (recall that there are only two cases: $I=0$ or $I>0$. NOT ...
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 ...
5
votes
1answer
58 views

$xf''(x) , xf', f \in L^{2}$ is $f' \in L^{1}$?

I am stuck on the following problem. I have a function $f$ such that $f$ is bounded on $(0,1)$, $xf'(x)$ is bounded on $(0,1)$, $f \in L^{2}(0,1)$, $xf' \in L^{2}(0,1)$, and $xf'' \in L^{2}(0,1)$. ...
1
vote
1answer
32 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], ...
-1
votes
2answers
35 views

Convergence of running maximum of uniform random variables [closed]

Let $X_1, X_2, ... X_n$ be an IID sequence of IID random variables that have a uniform distribution $(0,1)$. Let Max$(n) =$ max$(X_k:1\le k \le n)$, where $n\in \mathbb N$. How do I show that ...
3
votes
1answer
44 views

Proving a specific limit of integrals without using the Monotone Convergence Theorem

I am trying to prove the follow exercise without using the Monotone Convergence Theorem. Let $(X, \mathfrak{M}, \mu)$ be a measure space. Suppose $f \geq 0$ is measurable. Prove that $$\lim_{n ...
3
votes
1answer
108 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
1answer
43 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 ...
3
votes
0answers
35 views

Variation of a strongly bounded measure is strongly bounded too

Let $\mathcal{A}$ be a field of subsets of a set $\Omega$, $X$ a Banach space and $\mu:\mathcal{A}\rightarrow X$ a finitely additive vector measure. The variation of $\mu$ is the extended ...
1
vote
2answers
30 views

$\sigma$-finite measures and a sequence of simple measurable functions.

Let $(X, \mathfrak{M}, \mu)$ be a measure space. Suppose that $f$ is a non-negative, measurable function and that $\mu$ is $\sigma$-finite. Show that there exists a sequence $(\phi_{n})$ of simple ...
2
votes
2answers
54 views

If two Borel measures coincide on all open sets, are they equal?

Let $X$ be a topological space and let $\mathcal{B}(X)$ be its Borel $\sigma$-algebra. That is, $\mathcal{B}(X)$ is the smallest $\sigma$-algebra on $X$ containing all the open sets. Let $\mu, \eta : ...
2
votes
0answers
36 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
33 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
1answer
81 views

Usual convex combination and the one with measure

Let $X$ be a Borel measurable subset of $\Bbb R^n$ and let $\nu$ be a probability measure on $X$. Can we always find an integer $m$, points $x_1,\dots,x_m\in X$ and coefficients $a_1,\dots,a_m \geq ...
3
votes
0answers
46 views

Hopf lemma for generalized normal derivatives

Let $\Omega\subset\mathbb{R}^N$ be a bounded smooth domain. If $u\in C^2(\Omega)\cap C_0(\overline{\Omega})$ is a superharmonic function ($-\Delta u\ge 0)$ then, Hopf Boundary Lemma does implies that ...
1
vote
0answers
14 views

Can we identify Fourier transform of continuous compacltly supported functions with finte complex Borel measure?

It is well-known that, $L^{1}(\mathbb R)$ can be embed into $M(\mathbb R)$ (= The space of complex Borel measure on $\mathbb R$); by identifying $f\in L^{1}(\mathbb R)$ with the measure $d\mu= f dm.$ ...
3
votes
0answers
115 views

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 ...
3
votes
1answer
73 views

Extensions of universal measures

Let $(\Omega,\mathcal F)$ be a measurable space, and let $\mathcal P$ be the set of all probability measures no this space. Let $\mathcal F^p$ denote a completion of $\mathcal F$ w.r.t. $p\in P$ and ...
1
vote
1answer
26 views

Don't understand a $L^\infty$ bound argument involving measure of set

I'm trying to understand the proof of Proposition 2.2, part 2 of this paper. this is where I am stuck. For any $k > 0$, we have $$k^{\frac{2(N+1)}{N}}|\{|u|^m > k\}| \leq ...
1
vote
0answers
29 views

Measurability and composition of functions

If $f\circ p=h$ and $p:S^m \rightarrow S$ the $i{\text{th}}$ projection ($S$ measurable space) and $f:S\rightarrow \mathbb{R}$ any function whatsoever ($\mathbb{R}$=reals) and $h$ measurable, is there ...
1
vote
1answer
25 views

Question on $x$-section of measurable rectangle in product measure space $X \times Y$

I'm reviewing my analysis notes. We have that $(X, \Sigma, \mu)$ and $(Y, \tau, \nu)$ are complete measure spaces. We are considering the product measure space $(X \times Y, \Sigma(\lambda^{*}), ...
4
votes
2answers
55 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
2answers
36 views

For $1 \leq r < p < \infty$ prove the continuous injection of $L^p([0, 1])$ into $L^r([0, 1])$.

For $1 \leq r < p < \infty$ prove the continuous injection of $L^p([0, 1])$ into $L^r([0, 1])$. I am having a hard time starting. Any suggestions. I tried a straight forward approach. That ...
3
votes
0answers
41 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
46 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 ...
1
vote
1answer
40 views

Intersection of countable many sets of measure $1$

Consider a probability space $(X,\mathscr M,\mu)$ and a collection of measurable sets $\{A_n\}_{n\in\mathbb N}$ such that $\mu (A_n)=1$ for every $n$. Then I don't unterstand the following result: ...
2
votes
1answer
46 views

Find a non-negative function on [0,1] such that $t\cdot m(\{x:f(x) \geq t\}) \to 0$ that is not Lebesgue Integrable

Problem: Find a non-negative function $f$ on $[0,1]$ such that $$\lim_{t\to\infty} t\cdot m(\{x : f(x) \geq t\}) = 0,$$ but $f$ is not integrable, where $m$ is Lebesgue measure. My Attempt: Let ...
2
votes
1answer
42 views

When does intersection of measure 0 implies interior-disjointness?

If there are two "nice" shapes in $R^2$, such as circles or polygons, whose intersection has area 0, then they must be interior-disjoint, as their intersection can only contain pieces of their ...
3
votes
1answer
67 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
42 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 ...
3
votes
2answers
65 views

finite additivity&countable additivity

Let $\tau$ be a semialgebra of subsets of $\Omega$ and let P: $\tau\rightarrow [0,1]$, with $P(\Omega)=1$, and it satisfies finite additivity: $P\big(\bigcup_{i=1}^{n}D_i\big)=\sum_{i=1}^{n}P(D_i)$ ...
0
votes
2answers
26 views

Borel $\sigma$-algebra of subsets of [0,1]

Let sample space be [0,1]. let $\tau$ be the collections of all the open intervals on [0,1]. Borel $\sigma$-algebra is the smallest $\sigma$-algebra containing $\tau$. Is the statement any ...
1
vote
1answer
33 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 ...
1
vote
1answer
43 views

Space of Gaussian Functions is Closed in $L^2$

Let $\Omega, \mu$ be a probability space. A measurable function $f: \Omega \rightarrow \mathbb{R}$ is called Gaussian if $$\mu (f^{-1}(A))=\frac{1}{\sigma \sqrt{2\pi }}\int_Ae^{-x^2/2\sigma ^2} dx$$ ...
3
votes
1answer
32 views

countably additive function P

This problem comes from exercise 1.3.5(b) of 'A First Look at Rigorous Probability Theory'. It asks to give an example of a countably additive function $P$, defined on all subsets of $[0,1]$, which ...
3
votes
0answers
48 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 ...
4
votes
2answers
110 views

Evaluating $\lim_{n \rightarrow \infty} \int^{n}_{0} (1+\frac{x}{n})^{-n} \log(2+ \cos(\frac{x}{n})) \> dx$

The problem I am stuck on asks the reader to find the following limit: $$\lim_{n \rightarrow \infty} \int^{n}_{0} (1+\frac{x}{n})^{-n} \log(2+ \cos(\frac{x}{n})) \> dx.$$ The section I am working ...
0
votes
1answer
21 views

Finite union of measurable rectangles can be written as union of pairwise disjoint measurable rectangles?

Suppose we have two measure spaces $(X, \Sigma, \mu)$ and $(Y, \tau, \nu)$. $R \subseteq X \times Y$ is called a measurable rectangle if $R = A \times B$ with $A \in \Sigma$ and $B \in \tau$. I have ...