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

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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 ...
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2answers
23 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, ...
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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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1answer
52 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 ...
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2answers
37 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
21 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 ...
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1answer
20 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: ...
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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 ...
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32 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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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 $\mathbb{R}$?
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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 ...
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1answer
23 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 ...
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1answer
24 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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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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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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Jordan measure problem [on hold]

Any triangle is Jordan measurable,prove it.
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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.
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1answer
48 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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26 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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22 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
36 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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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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39 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
21 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
48 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 ...
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1answer
20 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, ...
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1answer
29 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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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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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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$\mu$ 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 $]0,1[ \not \in \sigma(\mathcal{S})$) ...
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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 ...
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1answer
20 views

Borel isomorphism and approximation of Borel space valued function

In Kallenberg's Foundations of modern probability, he defines a Borel space $(S,\mathcal{S})$ as a measurable space which is Borel isomorphic to a Borel subset $B\in\mathcal{B}([0,1])$, ie., there ...
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47 views

Fubini theorem counter example

While reading Kallenberg's proof of Fubini theorem in Foundations of modern probability, I realized that he first proved Tonelli's theorem, then apply Tonelli to $f_+$ and $f_-$, the positive and ...
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1answer
46 views

Why is the measure of a boundary of an open ball positive in only a countable number of cases?

Let $X$ be a Polish (complete separable metric) space and $\mathbb{P}$ a Borel probability measure on $X$. Let $x_1, x_2, \ldots$ be a sequence of points dense in $X$. How can you prove that there is ...
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35 views

Prove that $\int_c^d{f(y)dy} = \int_a^b{f(G(x))dG(x)}$

I'm doing this exercise from Real Analysis of Folland and got stuck on this problem. Let $G$ be a continuous increasing function on $[a, b]$ and let $G(a) = c, G(b) = d$. a) If $E ...
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2answers
47 views

Almost Everywhere Convergence versus Convergence in Measure

I am having some conceptual difficulties with almost everywhere (a.e.) convergence versus convergence in measure. Let $f_{n} : X \to Y$. In my mind, a sequence of measurable functions $\{ f_{n} \}$ ...
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Does a proper compact subset of a compact subset of R has strictly smaller measure?

Let K be a compact subset of R and H a proper compact subset of K. Does H has a strictly smaller Lebesgue measure than K?
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1answer
59 views

Use DCT to show:

$\lim_{n \rightarrow \infty} \int_0^{\infty}f_n(x)dx = \int_0^{\infty} \frac{x}{e^x-1}dx$, where $f_n(x):=\frac{n}{e^x-1}\sin\frac{x}{n}$ Hi I'm working on some practice questions and having a bit of ...
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Proving that a measure is continuous from below

Let $(X, \mathcal{M}, \mu)$ be a measure space and $\{E_j\}_{j=1}^\infty \subset \mathcal{M}$ such that $E_1 \subset E_2 \dots $ I want to prove that $\mu(\cup _1^\infty E_j) = \lim_{j \to ...
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Definition of $f \vee g$ and $f \wedge g$

In Olav Kallenberg's Foundations of Modern Probability he uses the notation $f \vee g$ and $f \wedge g$ where $f, g$ are two functions from a set $\Omega$ to $\mathbb{R}$. What does this notation ...
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If $\mathcal{B}$ is a base of a topology space $\left(X,\tau\right)$, then the Borel $\sigma$-algebra is generated by $\mathcal{B}$?

Let $\left(X,\tau\right)$ a topology space and $\mathcal{B}$ a base of the topology, my question is: The Borel $\sigma$-algebra is generated by $\mathcal{B}$ ?
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2answers
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What does the conditional expectation look like when the $\sigma$-algebra is infinite

Given a probability space $(\Omega,\cal F,\Bbb P)$, when $\sigma$-algebra $\cal F_0$$\subseteq \cal F$ is finite (which is generated by a finite partition $\Gamma \subseteq \cal F_0$), the conditional ...
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Application of Carathéodory outer measure theorem

We know Carathéodory outer measure theorem and its proof, but I want an application or example about this theorem (I mean give me an outer measure and show the class $Μ$) I don't want this outer ...
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Taking limit inside integration

What the conditions, other than DCT and MCT, under which $$\lim_{n\to\infty} \int f_n(x) \ \mathsf dx = \int lim_{n\to\infty} f_n(x) \ \mathsf dx\quad $$ where the $f_n$ are measurable functions? ...
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Is continuous random variable always an “onto function” [on hold]

Is continuous random variable always an "onto function"? If yes, why?
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Is probability mass function (PMF) the “law of X”?

Are they two the same? If not, what's the differences between these two? In continuous case, is PMF also equal to the integration of probability density function?
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Differentiability of parameter-dependent integrals when derivative exists only almost everywhere

This unanswered question asked in 2013 Differentiation under the Integral Sign (let's call this Q-zero) seems to be taken from this (or pdf ver.). The result on differentiation under the integral ...
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2answers
58 views

Is every $\sigma$-algebra generated by a partition?

I know that every finite $\sigma$-algebra is generated by a finite partition, but is every infinite $\sigma$-algebra also generated by "kind of" partition? Can anyone help provide a explanation or ...
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1answer
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Additive set function on a semiring of sets

A semiring $\Pi$ on a set $X$ is a non-empty family of subsets of $X$ with the following properties. 1) $P \cap Q \in \Pi$ whenever $P\in \Pi$ and $Q\in\Pi$. 2) $P - Q$ is a finite disjoint ...