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

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1answer
9 views

Equalities for the Upper and Lower Minkowski dimension definition

In a Geometric Measure Theory textbook the following was written: I cannot see how any of these equalities hold and dont believe they are obvious. If they are relatively obvious could someone ...
1
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1answer
19 views

Local convexity of the topology of weak convergence of probability measures

Let $X$ be a Polish or standard Borel space, and $\mathcal P(X)$ be the space of all Borel probability measures on $X$ endowed with the topology of weak convergence. I am thinking of using ...
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2answers
27 views

Are topologically well-behaved measure 0 subsets of $\Bbb R^2$ finite graphs?

Conjecture: If $X\subseteq \Bbb R^2$ is locally simply connected (hence locally path connected), compact and Lebesgue measure $0$ then $X$ is homeomorphic to a finite graph. It is clear that ...
3
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1answer
92 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 ...
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1answer
15 views

Properties of the Hausdorff measure

This comes from a book on geometric measure theory in a chapter introducing the Hausdorff measure $\mathcal{H^t}$. I cannot see in this proof how $\sum_i d(E_i)^s \leq \mathcal{H^s_{\delta}}(A)+1$ ...
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1answer
10 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 ...
1
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1answer
56 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 ...
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1answer
37 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 ...
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1answer
44 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. ...
5
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1answer
49 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 $xf'' \in L^{2}(0,1)$. ...
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1answer
44 views

$\lbrace D^+ f < c \rbrace$ is measurable when $f$ is continuous

Let $f: [a,b] \rightarrow \mathbb R$ be continuous and $$D^+f(x) = \limsup_{h \rightarrow 0^+} \dfrac{f(x+h) - f(x)}{h}.$$ Is the set $$\lbrace x \in [a,b] : D^+ f(x) < c \rbrace$$ Lebesgue ...
2
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1answer
37 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 ...
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1answer
21 views

Characterization of Sobolev Space

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
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0answers
39 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 ...
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0answers
30 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 ...
3
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1answer
36 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
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2answers
67 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 ...
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1answer
30 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 ...
2
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0answers
23 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, ...
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1answer
607 views

Example of Converge in measure, but not converge point-wise a.e.?

Can anyone give an exam of Converge in measure, but not converge point-wise a.e.? And also for the converse part, professor asks us to prove "pointwise a.e. implies converge in measure", but think ...
1
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1answer
27 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], ...
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votes
2answers
26 views

Convergence of running maximum of uniform random variables [on hold]

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
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0answers
27 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
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0answers
19 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 ...
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2answers
22 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
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2answers
46 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
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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 ...
2
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0answers
38 views

Usual linear 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 ...
1
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1answer
30 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
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2answers
50 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?
2
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0answers
18 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 ...
0
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0answers
11 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 $d\mu= f dm.$ ...
2
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0answers
28 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 ...
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0answers
10 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
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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 ...
5
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1answer
89 views

$E \subseteq [0, 1]$, $m(E) > 0$. Show that there are $\alpha$ and $\beta$ such that $\alpha, \alpha + \beta, \alpha + 2\beta \in E$.

This was originally a proof verification question, but I have since moved the proof to an answer as discussed on meta. I still welcome comments on the proof as well as any alternative proofs. ...
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1answer
57 views

Weak convergence of continuous functions

Let $X$ be an LCH space and $C_0(X)$ the set of continuous vanishing functions on $X$. If $C_0(X)$ is given the structure of a Banach space with the sup-norm, then its weak topology is given by the ...
3
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1answer
65 views

Is Steinhaus theorem ever used in topological groups?

Steinhaus theorem in $\mathbb{R}^d$ says that for $E\subset\mathbb{R}^d$ with positive measure, $E-E:=\{x-y:x,y\in E\}$ contains an open neighborhood of the origin. And for locally compact Hausdorff ...
3
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2answers
28 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
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2answers
52 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: $P\big(\bigcup_{i=1}^{n}D_i\big)=\sum_{i=1}^{n}P(D_i)$ ...
2
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1answer
29 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?
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0answers
57 views

Is this distributional laplacean a measure?

Let $\Omega\subset\mathbb{R}^N$ be a bounded smooth domain. Suppose that $0<k<t<1$ and $\Phi$ is a mooth function satisfying: $\Phi(x)=0$ for $x\le k$, $\Phi(x)=1$ for $x\ge t$. Take $u\in ...
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0answers
26 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 ...
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1answer
19 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^{*}), ...
3
votes
1answer
28 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 ...
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0answers
39 views

von Neumann Algebras and measures

I read that any abelian von Neumann algebra is isomorphic to $L^\infty(X,\mu)$ for some $X$ and $\mu$. This seems to be reasons, to consider any von Neumann Algebra as non-commutative measurable ...
2
votes
1answer
41 views

Unclear inequality in the proof of Birkhoff ergodic theorem.

I'm trying to understand the tricky proof of the ergodic theorem (Birkhoff 1931). My reference is "Ward,Einsiedler - Ergodic theory (with a view towards number Theory)" section 1.6: Consider the ...
7
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0answers
76 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))} < ...
1
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0answers
52 views

Why $\int f_0 \mbox{d}(F_0+F_1)=\int f_0 \mbox{d}F_0+\int f_0 \mbox{d} F_1=1$ should be true?

A measure $\mu$ dominates another measure $\nu$ whenever $\mu=0$ implies $\nu=0$. If I would like to take the integral of a measurable function $f_0$, say the density function of the probability ...
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 ...