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

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1answer
25 views

Prove that set of measure zero cannot have measure theoretic density away from 1

I've been trying to solve the following question from Bass: Fix $\epsilon \in (0,1).$ If a Lebesgue measurable set $A$ is such that for every bounded interval $I$, $m(A \cap I) \leq (1-\epsilon) ...
1
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0answers
10 views

bounding the measure of a disjoint union under a product measure

Let $(X\times Y,\mathcal{F},P)$ be a probability space such that $P$ is a product measure on $X\times Y$. Let $E=\cup_{x\in X}E_x$ be an event such that: (i) for each $x\in X$, $E_x$ is an event of ...
2
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0answers
29 views

Two different definitions of General Measurable Function

I've noticed two different kinds of definitions for a Measurable Function. In Folland's Real Analysis Modern Techniques: If $(X, \mathcal {M})$ and $(Y, \mathcal {N})$ are measurable spaces, a ...
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0answers
24 views

Is there any solution manual to Halmos' Measure Theory?

I've spent some time on Halmos' Measure Theory and must upvote such a good book. I want to solve most exercises in this book. I'm not sure whether there is a solution manual or instructor manual that ...
2
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0answers
29 views

Set $E\subset \mathbb{R}^n$ of positive Lebesgue measure such that the Lebesgue measure of $\bar{E}\backslash Int(E)$ is zero

Let $E\subset \mathbb{R}^n$ have positive Lebesgue measure. What are easily interpretable sufficient conditions on $E$ to guarantee that the difference between the closure $\bar{E}$ and the interior ...
8
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1answer
55 views

Will the Lebesgue integral of a real valued function always be a Riemann sum?

If we have a real valued integral that is Lebesgue integrable but not Riemann integrable, can the value of the Lebesgue integral be given by a Riemann sum by choosing appropriate points in the ...
0
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1answer
26 views

Example of uncountable union of sets that is in $\sigma$-algebra but does not satisfy additivity property of probability

Let $(\Omega, \mathcal{F}, \mathrm{P})$ be a probability space. I know that $\mathrm{P}$ should satisfy: $$ \mathrm{P}(\cup_{n \geq 1}A_n) = \sum_{n \geq 1} \mathrm{P}(A_n) $$ for disjoint family of ...
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0answers
28 views

Could two different sums yield two different areas?

I was talking with a professor and from what I understood: If you take a polygon and split it into various pieces: $$P=p_1+p_2+\dots + p_n$$ One needs to prove that the order of the sum does not ...
6
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1answer
59 views

A Vitali set is non-measurable, direct proof, without using countable additivity

I am teaching a measure theory class, where we are in the process of constructing Lebesgue measure on $\mathbb{R}$ via the usual Caratheodory outer measure construction. As motivation, we began by ...
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5answers
748 views

Why countable unions, intersections etc.?

I was just wondering why one always insists on countability when it comes to the definition of a $\sigma$-algebra in measure theory. I mean, measure theory works as it does, but is there a deeper ...
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2answers
44 views

Exercise: signed measures, total variation.

I have this exercise: Let $\nu_1$ and $\nu_2$ be finite signed measures on $(\Omega,\mathcal{A})$. Prove that: $|\nu_1+\nu_2|\le|\nu_1|+|\nu_2|$; , that is ...
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0answers
36 views

Question: outer measure

To define the outer measure of an arbitrary subset $E\subset \mathbb{R}^n$, cover $E$ by a countable collection $S$ of intervals $I_k$, and let $$\sigma(S) = \sum_{I_k \in S} v(I_k)$$ where $v(I)$ is ...
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1answer
21 views

Quick question about algebras

One of the requirements for an algebra $\mathcal{A}$ is that $\Omega$$\in$$\mathcal{A}$. But does $\Omega$ have to be it's own element in $\mathcal{A}$ or is it enough to let $\mathcal{A}$ consist of ...
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0answers
33 views

About the proof that step functions are dense in $L^p$.

I have some questions regarding step functions while reading a proof: If $f$ is in $L^p$ for $1\leq p <\infty$, then why is $f_n=\min\{2^{-n}\lfloor 2^n f\rfloor, n\}$ a simple function? ...
2
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2answers
29 views

A conjecture about generating algebras on a probability space

Suppose that $(X,\mathscr F,\mathbb P)$ is a probability space. Let $\mathscr E\subseteq\mathscr F$ be an algebra (i.e., it is a non-empty collection closed under complementation and finite unions) ...
0
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0answers
23 views

Let $(A_n)$ be a decreasing sequence of sets such that $A_n\to A=\cap_n A_n$, is the following true when measure is infinite? [duplicate]

Let $(A_n)$ be a decreasing sequence of sets such that $A_n\to A=\bigcap_n A_n$, is $$\mu \left(\bigcap_{n=1}^\infty A_n\right)=\lim_n \mu (A_n)$$ true when the measure $\mu (A_k)$ is infinite for all ...
3
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1answer
60 views

On the integration of a Lebesgue measurable function

Consider a function $f$ defined as $f:[0,2\pi]\to \mathbb{R}$ such that $\begin{equation} f(x)=\inf_{n\in \mathcal{N}} \sin^2 (2^n x) \end {equation}$ Is possible to give a decent bound of ...
1
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1answer
30 views

Characteristic function of a measurable set.

Let $X=L^p[0,1]$ $(1\leq p<\infty)$ be the Lebesgue space of p-integrable real functions on $[0,1]$. Let $D\subseteq [0,1]$ be measurable subset. The characteristic functions $\chi_D$ is defined as ...
1
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0answers
23 views

A small doubt regarding a previously asked limit of convolution

Previously, I asked this question to the forum. Pointwise limit of convolution Now, a question in this regard is coming to my mind. Suppose, we don't have the integral; i.e. we have the ...
4
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2answers
63 views

Showing that $(f_1,f_2,\dots,f_m)$ is a measurable function from $(\mathbb R^m,\mathcal B(\mathbb R^m))$ into itself

If $\{f_i, 1 \le i \le m\}$ is a set of real valued Borel functions on $\mathbb R$, how to show a vector of functions, $(f_1, f_2,..., f_m)$ is a measurable mapping from $(\mathbb R^m, ...
3
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1answer
40 views

If $\lambda=$ measure of a set and all $G_k$'s are open sets, then : $\lambda ( \cup_{k=1}^{\infty} G_k ) \le \sum _{k=1}^{\infty}\lambda ( G_k)$

I just started reading the book Lebesgue Integration on Euclidean Spaces by Frank jones, in which the author gives a result and it's proof as : the If $\lambda$ denotes the measure of a set and all ...
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2answers
79 views

Folland, Real Analysis Theorem 1.19

Theorem: If $E\subset\mathbb{R}$, the following are equivalent a.) $E\in M_\mu$ b.) $E = V\setminus N_1$ where $V$ is a $G_\delta$ set and $\mu(N_1) = 0$ c.) $E = H\cup N_2$ where $H$ is a ...
3
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2answers
43 views

Borel sets: alternative characterization for metric space

For any topological space $(X,\tau)$, the Borel $\sigma$-algebra $\mathcal{B}$ is the $\sigma$-algebra generated by the open sets. In other words, it is the intersection of all $\sigma$-algebras on ...
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2answers
29 views

Equivalence of Definitions of lim inf of Sequence of Sets

Prove : $\{w : w \in A_n \text{ for all $n$ except a finite number}\}= \bigcup_{n=1}^{\infty}\bigcap_{k=n}^{\infty}A_k$. I am trying to prove these two definitions are equivalent but I am having ...
3
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1answer
23 views

All closed rational rays measurable implies $f$ measurable

Is the following proof correct? Let $f: X \to \mathbb{R}$ where $X$ is a measurable space. Suppose $\{x: f(x) \geq r\}$ is measurable for each $r \in \mathbb{Q}$. Then, $f$ is measurable. Proof: ...
1
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1answer
8 views

Question about proof of finite additivity for measure on rectangles

I am trying to understand the proof of proposition 5.11 from these notes. Given two measure spaces $(X, \Sigma_X, \mu)$ and $(Y, \Sigma_Y, \nu)$, define a measure $\lambda$ on $(X \times Y, \Sigma_X ...
2
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1answer
45 views

Difficulty in understanding converse part of proof of a propostion in Andrew Browder's Mathematical Analysis

Proposition: Let $\mu$ be finitely additive set function, defined on the algebra $\mathscr A$. Then $\mu$ is countably additive if and only if its has following property: if $A_n \in \mathscr A$ ...
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0answers
34 views

Four definitions for Borel algebra in $\mathbb{R}$? [on hold]

Let us take $X=\mathbb{R}$, the set of real numbers. Of course we know a Borel algebra in $\mathbb{R}$. How can we have four definitions for a Borel algebra in $\mathbb{R}$?
2
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0answers
19 views

Measure-preserving map between a function and its symmetric rearrangement

Let $f \, \colon \mathbb{R}^d \rightarrow[0, \infty)$ be a function such that the sets $ \{ y \: \colon f(y) > \lambda \}$ are of finite Lebesgue-measure, for every $\lambda \geq 0$. Then, we can ...
1
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1answer
24 views

Quick question on why two measures have equal total mass.

I am following Probability with Martingales by Williams I am having troubles with why the two measures $H \rightarrow P(I \cap H)$ and $H \rightarrow P(I)P(H)$ have the same total mass $P(I)$. Is ...
0
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1answer
26 views

Approximation of characteristic function by mollifiers

I have been asked to show that the Heaviside function $H := \chi_{[0,+ \infty)}$ does not admit weak derivative in $L^1_{loc}(\mathbb{R})$. Here's my reasoning: By definition the weak derivative of ...
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0answers
17 views

Folland exercise 1.32

Here is a problems after the measure theory section. Suppose {$\alpha_j$} $\subset (0,1)$. a. $\prod $(1-$\alpha_j$) > 0 iff $\sum \alpha_j < \infty $. (Compare $\sum log(1- \alpha_j) to ...
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0answers
12 views

An elementary inequality in the context of Strictly Convex Function

Suppose: whenever $\epsilon \gt 0$ , define: $\zeta (y, \epsilon) = \frac{\eta (y+\epsilon) - \eta (y)}{\epsilon} - \eta'_{+}(y)$ ; where: $\eta$ is STRICTLY CONVEX CONTINUOUS FUNCTION & ...
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0answers
22 views

Jordan measure problem [on hold]

Any triangle is Jordan measurable,prove it.
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0answers
16 views

Measure theory problems on elementary set [on hold]

If $E$ and $F$ are elementary sets in $\mathbb{R^n}$,then $E\cup F$,$E\cap F$ are also elementary.
1
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1answer
53 views

Show $F: (0, \infty) \rightarrow \mathbb{R}$ is diff'ble

Hi just need a bit of help with a few parts of this practice question: Show $F: (0, \infty) \rightarrow \mathbb{R}$ diff'ble with respect to $t \in (0, \infty)$, where $$F(t) := ...
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0answers
31 views

Example where probability theory fails without $\sigma$-algebra

I have just started reading theory of probability in a measured theory based approach and was wondering if someone could give an example where probability fails without using $\sigma$-algebra (or ...
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0answers
24 views

Looking for a bounded set in a set with finite measure lebesgue.

Let $A\subseteq\mathbb{R}^{n}$ with $\mu^{*}\left(A\right)<\infty$. Show that for each $\varepsilon >0$ there is $A_{\varepsilon}\subseteq\mathbb{R}^{n}$ bounded such that ...
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1answer
38 views

Algebra that is not a $\sigma$-algebra

Let $X=\Bbb R\ $ and$\ $ $\mathcal A=\{\text{finite disjoint unions of}\ (-\infty,b],\ (a,b]\land(a,\infty)\}$. So the exercise says to prove that $\mathcal A$ is an algebra but is not a ...
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0answers
18 views

Orderings on a space such that every initial segment has measure 0

Let $(\mu,X,\Sigma)$ be an atomless probability measure. Is it alway possible to find a well-ordering of $X$, $<$, such that for any $x\in X$, $Pr(\{y\mid y<x\})=0$? (Edit: I'd also be ...
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2answers
46 views

Show that for every $\epsilon > 0 $ there exists $h \in \mathcal{L}^1(X)$ non-negative and $\delta > 0$ such that:

I am working through some practice questions, and I think I have gotten the first two parts, but I am having trouble deriving the third part: Let $(X,\mathcal{A},\mu)$ be a finite measure space. ...
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1answer
23 views

On a proof regarding the sigma algebra generated by a single random variable.

I left (b) and (c) for the sake of the curious. What I am trying to do is Exercise (a) except that I recall that $\sigma(Y):= ( \{ w : Y(w) \in B \} : B \in \mathcal{B} )$ is the definition of ...
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3answers
49 views

Difference between convergence in measure and convergence almost everywhere

This question is an extension of a question asked earlier. Let $(X,\mathcal{M},\mu)$ be a measure space and let $f_{n}: X \to Y$, where $\{f_{n}\}$ is a sequence of functions. The proof wiki ...
3
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1answer
23 views

Can this be proved using the MCT instead of the DCT?

I've seen various version of the DCT prove that if $f$ is a real valued, or extended real valued, or complex, integrable function, and if $\{E_n\}_n$ is a sequence of disjoint measurable subsets, ...
2
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1answer
38 views

Pointwise limit of convolution

Suppose $\omega$ is the standard mollifier in $\mathbb R$. Then, let $\omega_{\epsilon} (x):= \frac{1}{\epsilon} \omega \left(\frac{x}{\epsilon}\right)$. For $0 < t_{1} < t_{2}$ the following ...
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0answers
22 views

Explicit construction of Haar measure on a locally profinite group

Let $G$ be a locally profinite group. A Haar measure $\mu$ on $G$ is a measure defined on the $\sigma$-algebra $\mathcal B(G)$ of all Borel sets of $G$ with the following properties. 1) $\mu(K) ...
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0answers
36 views

Counterexample of the failure of integration by parts when we only assume differentiability of $f$ and $g$.

Counterexample of the failure of integration by parts when we only assume differentiability of $f$ and $g$. We know when $f$ and $g$ are both AC functions, the integration by parts is true. Is it ...
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0answers
21 views

$\mu$ is a finite Borel measure on $\Bbb R$, absolutely continuous w.r.t. to the Lebesgue measure $m$. Prove that $x \mapsto \mu(A+x)$ is continuous.

Let $\mu$ be a finite Borel measure on $\Bbb R$, which is absolutely continuous with respect to the Lebesgue measure $m$. Prove that $x \mapsto \mu(A+x)$ is continuous for every Borel set $A \subseteq ...
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1answer
16 views

Borel $\sigma$-algebra and natural number

Let $\Omega=\mathbb{R} $ and $\mathcal{S}=\{\{x\}:x \in \Omega \}$ a) $\mathbb{N} \in \sigma(\mathcal{S})$? b)Prove that $]0,1[ \not \in \sigma(\mathcal{S})$) ...
3
votes
2answers
45 views

Applying the definition of Lebesgue Integral to specific functions

I am fairly sure this question will sound rather naive, but I do have a problem with applying the Lebesgue Integral. Actually this question can be divide in two sub-question, related to two examples I ...