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

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12 views

About a countably additive vectorial measure

I need a hand with this exercise. We consider the space $([0,1],\cal{B},\mu)$, where $\mu$ is the Lebesgue measure. Let $1\leq p< \infty$. We define $F:[0,1]\to \ell^p$ as ...
7
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3answers
2k views

If $S$ is an infinite $\sigma$ algebra on $X$ then $S$ is not countable

I am going over a tutorial in my real analysis course and there is an exercise that I don't understand a part of his solution. The exercise is: Let $S$ be an infinite $\sigma$ algebra on $X$ .Prove ...
1
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1answer
39 views

How does Wikipedia's definition of the Lebesgue integral relate to more common definitions?

Wikipedia presents a definition of the Lebesgue integral (of a nonnegative function $f$) that I hadn't encountered before: Let $f^*(t)=\mu \left (\{x\mid f(x)>t\} \right )$. The Lebesgue ...
2
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0answers
15 views

Problem with infinite product measures

Given some measurable space $\left(X,\mathcal{F}\right)$ and two probability measures $\mu$ and $\nu$ on this space one can define ...
0
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0answers
11 views

prove that a simple function on X is measurable wrt M iff Ai are all elements of M. (full question in description)

let X be a set and M be a sigma-algebra in X. (a) prove that a simple function $s = \sum_{i=1}^{n} c_i\chi_{A_i}$ on X, where the {$c_i$} are distinct and not zero, is measurable wrt M iff $A_i$ are ...
0
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1answer
13 views

Let $X$ be any set and $M=\{\emptyset, X\}$. Prove that the class of measurable functions are exactly those functions that are constant on X.

I am attempting to solve a suggested problem while studying for my upcoming real analysis exam. Could somebody please help me with this question? Question: Let $X$ be any set and $M=\{\emptyset , ...
8
votes
2answers
829 views

$C_c(X)$ dense in $L_1(X)$

I did the following homework, can you tell me if my answer is correct? Prove that for any $\sigma$-compact, locally compact Hausdorff space $\Omega$ equipped with a Radon measure $\mu$ the set of ...
0
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1answer
33 views

Support of $L^p$ functions?

I noticed something strange. If we look at a function $f \in L^p$, then this is an equivalence class. Hypothetically: $\operatorname{supp}(f) = \overline{\{f\neq 0\}}$. But this is strange, as $f$ is ...
0
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0answers
18 views

If $f_{n} \in L^{\infty}$, $ \int_{0}^{1}f_{n_{k}}(x)g(x)dx \rightarrow \int_{0}^{1}f(x)g(x)dx$ for every $g \in L^1$

Supposet that $\{f_{n}\}_{n=1}^{\infty} \in L^{\infty}$. Is the following statement always true? There is a subsequence $\{n_{k}\}$ and a function $f \in L^{\infty}$ such that $$ ...
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1answer
47 views

equi integrablity

See page 5 here. Let $\Omega$ be an open subset of $\mathbb{R}^n$, and let $(f_n)$ be a sequence of measurable functions, $f_n \in L^1(\Omega)$, which is bounded in $L^1(\Omega)$ ($f_n \in ...
1
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1answer
66 views

convergence in L^{1} strong

I search an proof of this lemma: First,we have this definition: we tell that an sequence $f^{\epsilon}$ is equi integrable if $$\forall \eta, \exists \delta > 0, |E|\leq \delta \implies ...
0
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1answer
17 views

Lebesgue measure unique on semiring?

in our lecture it was stated that the Lebesgue measure can be uniquely extended from a semiring to a sigma algebra by Caratheodory's theorem. Unfortunately, we did not show that it is unique on the ...
1
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1answer
29 views

the preimage of a continuous function is measurable?

Suppose that $f(x)=|x|$ in $R^d$, then can we show that for any measurable subset $E\subset R^1$, $f^{-1}(E)$ is measurable in $R^d$? P.S. It should be noted that this is not true for general $f$ ...
3
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2answers
165 views

Integration problem, existence of a constant in a inclusion between integrable functions and essentially bounded functions

Let $\mu$ be a finite positive measure on the borel sets in $\mathbb{R}$ and suppose $\mathcal{L}^{1}(\mathbb{R},\mu) \subset \mathcal{L}^{\infty}(\mathbb{R},\mu)$. Show that there exists $c>0$ ...
2
votes
1answer
35 views

Estimate for weak $L^{1}$ norm

Let the weak $L^{1}$ norm on $f$ be defined by $\|f\|_{\mathrm{WL}^{1}} = \sup_{t > 0}t D_{f}(t)$ where $D_{f}(t) = \mu(\{x \in \mathbb{R}: |f(x)| > t\})$ and $\mu$ is the standard Lebesgue ...
1
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1answer
43 views

Prove that volume of a ball in a polytope is very small

An exercise in a book asks to prove that for a bounded convex polytope $P\subseteq\mathbb{R}^n$ defined as an intersection of $k$ closed halfspaces and for a unit ball $B^n$ contained in $P$ the ...
0
votes
1answer
22 views

Show that the image of a zero measure set is of zero measure

I saw a topic on the subject but I did not quite understand, and it was a bit old and I didn't want to resurrect it. I am going in the right direction, I just need a little nudge. let $f: \mathbb ...
1
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1answer
19 views

Proof about finitely additive measure

Let $X$ be some set, and let $\Sigma$ be a $\sigma$-algebra on $X$. Assume that $\mu:\Sigma\to[0,\infty]$ is a finitely additive measure on $\Sigma$, that is $\mu(\emptyset)=0$ $U\cap V=\emptyset ...
13
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2answers
3k views

Limit of $L^p$ norm

Could someone help me prove that given a measure space $(X, \mathcal{M}, \sigma)$ and a measurable function $f:X\to\mathbb{R}$ in $L^{\infty}$ and some $L^{q}$, ...
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1answer
99 views

Let $\{f_n\}_{n=1}^\infty$ be non-negative functions and $f_n \to f$ then $f \geq 0$

I have trouble with this question: Let $\{f_n\}_{n=1}^\infty$ be a sequence of non-negative functions in $L^2(0, 1)$, and suppose that $f_n$ converges to a function $f$ in the norm of $L^2(0, 1)$. ...
0
votes
2answers
76 views

Let $D$ be a subset of $L^2[0,1]$ defined in the following way

any help with this problem, it gave me hard time: Let $D$ be a subset of $L^2[0,1]$ defined in the following way: A function $f$ belongs to $D$ if and only if f is equal almost everywhere to a ...
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1answer
57 views

$\min \{||x-x_0|| : x \in V \} = \max \{|<x_0,y> : y \in V^{\bot} \}$

I need some help, please Let $V$ be a closed subspace of Hilbert space $H$, and let $x_0 \in H$. Show that $\min \{||x-x_0|| : x \in V \} = \max \{<x_0,y> : y \in V^{\bot} \}$ thanx in ...
0
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1answer
60 views

there is a measurable function $f$ on $X$ such that $|{f(x)}|=1$ for a.a $x \in X$ and $\nu(E)=\int_Efd|{\nu}|$ for any $E \in \mathfrak{M}$

any hints on this problem: Let $\nu$ be a finite signed measure on a measure space $(X, \mathfrak{M})$ and let $|{\nu}|$ be its total variation, prove that there is a measurable function $f$ on $X$ ...
0
votes
3answers
107 views

Hi I was thinking about a problem and have a question: [on hold]

Hi I was thinking about a problem and have a question: we know that if $f∈C([0,1])$ for which $∫_0^1f(x)x^ndx=0$ for all $n \geq 0$ then $f=0$! Now my question is: Do we still have the same when we ...
0
votes
0answers
11 views

Is there a relationship between Haar measures and Haar wavelets, other than the name Haar?

Is there a relationship between Haar measures and Haar wavelets? The Haar wavelet does not seem to be invariant to translations, nor rotations.
1
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1answer
56 views

weak convergence of a bounded linear operator [duplicate]

I need help with this problem Let $X$ be a reflexive Banach space and $T: X \to X$ a linear operator. Show that $T$ belongs to $\mathcal{L}(X,X)$ if and only if whenever $\{x_n \}$ converges weakly ...
4
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1answer
56 views

Trouble finishing a (direct) proof that $\ell^2(A)$ is a complete metric space

Let $A$ be any non-empty set. We can define summations of non-negative numbers over this index set by using a supremum of summations over finite subsets of $A$. That is, $$\sum\limits_{\alpha \in A} ...
1
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2answers
58 views

What does this set look like? [on hold]

Consider the set of points $A \subset \mathbb{Q}^2$ in the unit square with both coordinates rational: $$A=\{(x,y) \in \mathbb{Q^2} \mid \, 0 \leq x \leq 1,\,\, 0 \leq y \leq 1 \} $$ If we color the ...
2
votes
1answer
30 views

$L^{2}$ integrability implies $L^{1}$ integrability on sets of finite measure.

Let $X$ be a measurable space with $m(X) < +\infty$. I think it's clear that if $f \in L^{2}(X)$ implies that $f \in L^{1}(X)$. But when $m(X) = +\infty$, the suppose $f(x) = \frac{1}{1+|x|}$ is ...
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votes
0answers
24 views

change of variable in the integral with measure $\mu$ [on hold]

Let $f:(X,\mathbb{X}, \mu) \to Y$ a measurable function and $\nu(A)=\mu(f^{-1}(A))$. show that $$\int_Yg \,d\nu = \int_X g(f(x)) \,d\mu(x)$$
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0answers
12 views

Clarification on set notation for set of points where a given sequence converges.

Prove that, given a sequence of measurable functions $\{f_{n}\}$, the set of points at which $\{f_{n}\}$ converge is measurable. My solution is to first define $f(x) = \limsup_{n \to \infty} ...
0
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0answers
27 views

Generate Borel Sigma Algebra

I want to show that the Borel Sigma-Algebra on $\mathbb{R}^n$ is generated by $ A:= \{(a_1,b_1] \times \cdots\times (a_n,b_n]; a_i,b_i \in \mathbb{R} \}$ as well as $ B:= \{(-\infty,c_1] ...
3
votes
1answer
28 views

Suppose a collection of unit vectors has measure zero on the sphere. Can ${\mathbb R}^d$ be the union of the subspaces perpendicular to the vectors?

So if a union of proper subspaces has measure zero (e.g. countably many subspacees), then ${\mathbb R}^{d}$ is not the union of these proper subspaces. But what if we have a union of $d-1$ dimensional ...
0
votes
1answer
40 views

Meaning of symbol $L^1(\mathbb{P})$

In Furstenberg-Kesten theorem, a theory relating to products of random matrices, one of the assumptions is that: $$\log^{+}||A||\in L^1(\mathbb{P}),$$ where $A$ (a random matrix) is the generator ...
3
votes
2answers
74 views

What is so special about the Lebesgue-Stieltjes measure

A measure $\lambda: B(\mathbb{R}^n) \rightarrow \overline{{\mathbb{R_{\ge 0}}}}$ that is associated with a monotone increasing and right-side continuous function $F$ is called a Lebesgue-Stieltjes ...
2
votes
0answers
33 views

Confused about the definitions of atom

I went though half a dozen books on measure theory, and it occurs to me that the definition of atom is not particularly unified. Version $1$: A set $E$ in a $\sigma$-algebra $\Sigma$ is called ...
0
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0answers
22 views

Showing an operator kernel is nowhere dense…

We are doing a problem that requires us to, for fixed NONZERO element $h \in L^2[0,1]$ (as in, the function $h$ is not zero on a set of positive measure in $[0,1]$) and fixed $f \in L^2[0,1]$ with ...
1
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1answer
24 views

A question on measures/densities

Let $Q,P,\nu$ be three measures on the same space such that $P\ll Q,P\ll\nu,Q\ll\nu$. Define $p=\frac{dP}{d\nu},q=\frac{dQ}{d\nu}$. Then $$\frac{1}{2}\int|p-q|d\nu=1-\int\min\{p,q\}d\nu$$ This should ...
2
votes
2answers
141 views

Showing that function is not Lebesgue Integrable in $[0,1]$

Is an exercise of my course of Measure and Integration. Let $f:[0,1]\rightarrow\mathbb{R}$ such that: $$ f(x)= \left\{ \begin{array}{ll} x^2\sin(\pi/x^2) & \textrm{ if } 0<x\leq 1\\ 0 ...
0
votes
1answer
9 views

non uniform convergence of integrable functions

Let $(f_n)\subseteq L(X,\mathbb{X},\mu)$ and $f_n\longrightarrow f$, then I must show that if $\lim_{n}\int \mid f_n-f\mid=0$ then $\int\mid f\mid d\mu =\lim_n \int\mid f_n\mid d\mu$. i don't know ...
0
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0answers
12 views

limitting value in p-norm

Let $f,g \in L^{p} ( \mathbb R^{n} , m)$ ; $m$ being the Lebesgue measure on $\mathbb R^{n}$ . Now , for $1\le p < \infty$ find the valoue of $lim_{|x|_{n} \to \infty} ||f(x+x_{n}) + ...
0
votes
1answer
22 views

Integration with respect to counting measure.

I am having trouble computing integration w.r.t. counting measure. Let $(\mathbb{N},\scr{P}(\mathbb{N}),\mu)$ be a measure space where $\mu$ is counting measure. Let ...
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0answers
15 views

How to show: $l^*(A)\le l^*(G_\epsilon)\le l^*(A)+\epsilon $ , $l^* $:outer measure

If $A$ s a Lebesgue measurable subset of $\mathbb{R}$ and $\epsilon\gt 0$ How to show: $\exists$ an open set $G_\epsilon \supset A$ such that $l^*(A)\le l^*(G_\epsilon)\le l^*(A)+\epsilon $, $l^* ...
3
votes
0answers
35 views

Why we are studying Dirac measure?

If we are doing mathematics for sth worthwhile not just for fun then why do we deed to work on Dirac measure $\delta_x$ at a point $x$? For an arbitrary measure space $(X,\scr{A},\delta_x)$, where ...
1
vote
1answer
14 views

Computation of integration wrt counting measure.

Let $f:\mathbb{R}\rightarrow\mathbb{R}$ is given by $f(x)=(x^2+3)\chi_{[0,2]}(x)$. Then how do we compute $\int_{\mathbb{R}}f\,d\mu$ where $\mu$ is counting measure?
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1answer
19 views

measurable function zero everywhere. [on hold]

Let $f:\Omega\rightarrow \mathbb{R}$ be measurable and assume that $\int_Af\,d\mu=0$ whenever $A\in\mathcal{F}$.Prove that $f=0$ almost everywhere.
0
votes
1answer
19 views

Bartle - integration, monotone convergence theorem

Suppose that $(f_n) \subset M^{+}(X, \mathbb{X})$, that $(f_n)$ converges to $f$, and that $\int f d\mu=\lim \int f_n d\mu < +\infty$. Prove that $$\int_E f d\mu=\lim \int_E f_n d\mu $$ for each ...
9
votes
0answers
205 views
+400

$I=\frac{1}{\pi}\int_0^{\pi/3}\log\big( \mu(\theta)+\sqrt{\mu^2(\theta)-1} \big)\ d\theta, \quad \mu(\theta)=\frac{1+2\cos\theta}{2}.$

Hi I am trying to calculate this integral: $$ I=\frac{1}{\pi}\int_0^{\pi/3}\log\left( \frac{1+2\cos\theta}{2}+\sqrt{\bigg( \frac{1+2\cos\theta}{2} \bigg)^2-1} \right)\ d\theta. $$ The ...
0
votes
1answer
12 views

sequence function question [on hold]

Let $(f_n)_{n=1}^\infty$ be a sequence in $\mathcal{R}[a,b]$ such that $f_n(t)\downarrow0$ for each $t\in[a,b]$. Does it follow that $\int_a^bf\rightarrow 0$? Proof or counterexample.
0
votes
1answer
24 views

Riemann integral differentiable

Let $\mathcal{f}:[a,b]\rightarrow\mathbb{R}$ be Riemann integrable, and define $F(t)=\int_a^tf$ for $t\in[a,b]$. Recall that $F$ must be continuous but need not be differentiable. Prove that $F$ is ...