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

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If $(f,g)$ is measurable than $f$ and $g$ are measurable?

Suppose $f : (Z,z) \rightarrow (X,x)$ and $g : (Z,z) \rightarrow (Y,y)$. If $(f,g)=(Z,z) \rightarrow (XxY, x⊗y)$ are measurable then $f$ and $g$ are measurable? If so how can I show that? My approach ...
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Countable Sets, sigma-algebra and probability measure

The problem I'm trying to solve is the following: Consider Ω uncountable and F = {A ⊆ Ω: A is countable or Ac is countable}. 1) Show that F is a σ-algebra of Ω 2) Consider P: F → [0,1]. If A is ...
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Generators of relative $\sigma$-algebras

Let $(X,\mathscr B)$ be a measurable space. For some $Y\subseteq X$, $Y\in\mathscr B$ and define $$\mathscr B_Y\equiv\{Y\cap B\,|\,B\in\mathscr B\}$$ to be the “relative $\sigma$-algebra” on $Y$ ...
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34 views

Stochastic dominance characterization

Consider two probability measures on $\Bbb R$ given by $\mu$ and $\nu$. We write $\mu\leq \nu$ if there exists a joint distribution $P$ with the latter marginals such that $P(x\leq y) = 1$. In ...
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Hausdorff measure of $f(A)$ where $f$ is a Holder continuous function.

Let $f\colon \mathbb R^d\to \mathbb R^k$ be a $\beta-$ Holder continuous function ($\beta \in (0,1)$) and $A\subset \mathbb R^d$. As for a Lipschitz function $g$ it holds that $H^s(g(A))\leq Lip(g)^s ...
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32 views

Time derivative of an invariant probability measure

Consider a dynamical system defined through a vector field $F$ in $M \subset \mathbb{R^n}$ that generates a flow $\Phi^t$ of the form $$\bf{\Phi^t X_0 = X} \ , $$ being $X_0 \in \mathbb{R}^n$ the ...
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Question on Product Spaces

Is it true that for $\sigma$-finite measure spaces $(X, \mathcal{F}, \mu)$, $(Y,\mathcal{G},\nu)$, if $f$ is a ($\mathcal{F} \times \mathcal{G}$)-measurable function on $X \times Y$, then $$\int ...
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What is the distribution function of a measure? (in general case)

Let $\mu$ be a measure on $\mathbb{R}_+$. What is the distribution function of $\mu$? (I want to understand distribution functions in general case) Thank you!
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$H^s(A)=0 \iff H^s_\infty(A)=0$

I have the following exercise: Show that $H^s(A)=0 \iff H^s_\infty(A)=0$ for $A\subset \mathbb R^d$. Here $H^s(A)$ is he Hausdorff measure of the set $A$, so $H^s(A):=\lim_{\delta\to 0^+} ...
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Dice Rolling Probability Space

I have a question in which the person asking has identified that the total sum of 11 comes up more often than a sum of 12 in the rolling of three dice and this is strange as they both have the same ...
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63 views

Hausdorff measure of rectifiable curve equal to its length

Let $(\mathbb{R}^n,d)$ be a metric space. A continuous, injective mapping $\gamma: [0,1]\to \mathbb{R}^n$ is a curve and denote its image $\overline{\gamma}:=\gamma([0,1])$. I wish to prove that its ...
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41 views

Find the minimal algebra and $σ$ algebra

Let $A$,$B$,$C$ be three arbitrary sets since they are all subsets of $Ω$. Find the minimal algebra and $σ$ algebra generated by these three sets. I did not understand this question. How can I ...
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51 views

Show that A is an algebra

Suppose $X$ is a collection of sets and $Ω$ element of $X$. Also $A$, $B$ are elements of $X$. Then, $A-B=A\cap B^c$ element in $X$. Show that $X$ is an algebra. Please help me, how I can show this ...
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Some Problems in the book ‘A course in Probability Theory’, K.L. Chung

Q1.(page10.6) A point $x$ is said to belong to the support of the d.f.(distribution funtion) $F$ iff for every $\epsilon>0$ we have $F(x+\epsilon)-F(x-\epsilon)>0$. The set of all such $x$ is ...
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79 views

A rectangle R is the union of a zero set and countably many disjoint discs

Prove that a rectangle R is the union of a zero set and countably many disjoint discs. This question is from Pugh's Real Mathematical Analysis, page $424$ question $39.b$. Any hint on proving this ...
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Can a continuous surjection between compacts behave bad wrt Borel fully supported probability?

Suppose A,B are compact metric spaces with fully supported Borel probability measures $m_A$ and $m_B$ correspondingly. Suppose that $A$ is second-countable. Let $f:A\to B$ be a continuous surjection. ...
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67 views

Beppo Levi's theorem

Let $f_n$ is a sequence of integrable functions and $\sup_n\int f_n d\mu<\infty$. I need to show that, if $f_n\uparrow f$, then $f$ is integrable and $\int f_nd\mu \rightarrow \int fd\mu$. This is ...
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37 views

Lebesgue measurability of $(0,1)$

I can show that any interval of the form $(a,\infty)$ is Lebesgue measurable from first principles. Thus $(0,\infty)$ and $$ (-\infty, ...
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37 views

Measure Theory: Continuity

How can I find a subset of a set with "half the size" of the original? I am trying to solve this problem and I came across this post. The solution using the intermediate value theorem is ...
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$(f_n)$ Borel measurable implies $\sup_n f_n$ and $\inf_n f_n$ Borel measurable

Suppose $(f_n)$ is a sequence of Borel measurable functions. Show that both $\sup_nf_n$ and $\inf_nf_n$ are Borel measurable. Attempt: Suppose $(f_n)$ is a sequence of Borel measurable ...
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22 views

Proof on the exterior measure

Let $E \subset \mathbb{R}^n$ be a measurable set and let $\delta > 0$. Show that $m^*(\delta E)=\delta^n m^*(E)$, where $m^*$ is the exterior measure (outer measure). Attempted proof: Let ...
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37 views

Completion of borel sigma algebra with respect to Lebesgue measure

There are two ways of extending the Borel $\sigma$-algebra on $\mathbb{R}^n$, $\mathcal{B}(\mathbb{R}^n)$, with respect to Lebesgue measure $\lambda$. One can take the completion ...
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66 views

“Basic” Proof of Riemann Lebesgue Lemma

Can someone provide a proof of the Riemann Lebesgue Lemma that doesn't make use of the known integral of $e^{itx}$? In particular I'm looking for the solution to the following problem: Show that ...
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23 views

Borel Sets and Borel Functions

I read the following argument in Billingsley (1995). Consider a function $F(x)$ and define four associated quantities at $x$: $$ D^F(x) := \limsup_{h\downarrow 0} \frac{F(x+h)-F(x)}{h}; \\ D_F(x) := ...
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Prove that $\mathscr{H}^0(F) = |F|$.

As stated above, I'd like to prove that the 0-dimensional Hausdorff Measure of a set $F \subset \mathbb{R}^n$ is the cardinality of $F$. In other words, that $\mathscr{H}^0 (F) = |F|$, or the number ...
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45 views

$\|f\|_2\le\|f\|_4^{\frac{2}{3}}\|f\|_1^{\frac{1}{3}}$

I want to prove that $$ \|f\|_2\le\|f\|_4^{\frac{2}{3}}\|f\|_1^{\frac{1}{3}} $$ I proved it by Holder inequality. But this is an exercise under "Interpolation". So I guess it can be proved using ...
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38 views

Dudley’s exercise on finitely additive probabilities

Yesterday I found the following exercise in Dudley’s “Real Analysis and Probability”. In a game, two players, Sam and Joe, each pick a nonnegative integer at random. For each, the probability that ...
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26 views

Closed Sets are Measurable Proof in Stein/Shakarchi Text

Had a question about the proof laid out in Stein/Shakarchi's text that closed sets are measurable. Im hoping someone can explain the part in bold below, which is the only thing that's bugging me. They ...
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Lebesgue Measurable functions closed under monotone limits

Let $f_n:D\to \mathbb{R}$ be measurable for all $n$ and suppose that $f_n(x)\leq f_{n+1}(x)$ for all $n$ and for all $x\in D$. If $\lim_{n\to\infty}f_n(x)=f(x)\forall x\in D$, show that $f$ is ...
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In Real Analysis by Royden, chapter 11, section 4 proposition 17 why the domain of integration is not mentioned.

In the above said text book chapter 11, section 4, proposition 17 why the domain of integration is not mentioned.
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Caratheodory: Infinite Extension

Given a $\sigma$-ring. Consider a premeasure on it: $$\mu:\mathcal{R}\to\overline{\mathbb{R}}_+:\quad\mu(R)=\sum_{k=1}^\infty\mu(R_k)\quad\left(R=\biguplus_{k=1}^\infty R_k\right)$$ Extend this to a ...
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Origins of measure and integration

Calculating integrals is often called as "Computing an area or a volume". More formal, integrals are measures defined in a certain space (satisfying $\sigma$-additivity and sometimes invariance ...
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Borel Measures: Integral Approximation

Problem Given a locally compact Hausdorff space. Consider a regular Borel measure in the sense: $$\mu:\mathcal{B}(\Omega)\to\overline{\mathbb{R}}_+:\quad\mu(E)=\inf_{E\subseteq ...
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42 views

Generating $\pi$-system for $\mathbb{Z}_+$

As we know from the definition a $\pi$-system is a collection where the intersection of two sets is again in that collection. In fact we can use $\pi$-systems to generate $\sigma$-algebra's. We know ...
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39 views

Showing that $\left\{ {\mathop {\lim }\limits_{n \to \infty } {X_n} = X} \right\}$ is an event

Let $X$ and ${\left( {{X_n}} \right)_{n \in \mathbb{N}}}$ be random variables on a measurable space $\left( {\Omega ,\mathcal{F}} \right)$. Show that: 1) $\left\{ {\omega \in \Omega :\mathop {\lim ...
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$A$ not Lebesgue-measurable in $\Bbb{R} \implies A \times \Bbb{R}$ is not Lebesgue-measurable in $\Bbb{R}^2$?

How do I show that if $A$ is not Lebesgue-measurable in $\Bbb{R}$ then $A \times \Bbb{R}$ is not Lebesgue-measurable in $\Bbb{R}^2$?
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Integral Existence

If we know that $$\int_{\mathbb{R}_+}\frac{d|\mu|(t)}{t+1}<\infty,$$ how can I prove that $$\int_{\mathbb{R}_+}\frac{z}{1+tz}d\mu(t)<\infty$$ where $z \in \mathbb{C}$, $\mu$ is a complex Radon ...
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Sigma finite measure generated by field

This is a question from Strichartz, The Way of Analysis (qu 15 in 15.1.4): If $\mu$ is a $\sigma$-finite measure on a $\sigma$-field $F_{1}$ generated by a field $F$, show that in the decomposition ...
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22 views

Uniform convergence on a measurable set implies convergence a.e.?

Suppose for each $\epsilon$ there exists a measurable set $F$ such that $\mu(F^c) < \epsilon$ and $f_n$ converges to $f$ uniformly on $F$. Prove that $f_n$ converges to $f$ a.e. I have been ...
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49 views

Check for integrability

Check as for the the integrability the following function $$g(x)=\frac{e^{-x^2}}{|x|^a}, a>0$$ How could we do this?? Do we have to check if $g \in L^1$. But at which space??
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Finding limit of integral

LEt $f:\Omega\to[0,\infty]$ measurable s.t $0<c:=\int_{\Omega}fd\mu<\infty$ . Prove that $$\lim_{n\to\infty}\int_{\Omega}n\log\bigg(1+\big(\frac f n\big)^a\bigg)d\mu=\cases{c\quad ...
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if $f:([0,1],P([0,1])) \rightarrow (\mathbb{R},\mathcal{B})$, what are the possible measurable functions?

if $f:([0,1],P([0,1])) \rightarrow (\mathbb{R},\mathcal{B})$ and $\mu$ is a measure which is defined as $\mu(A) = 0$ if $|A|\leq\mathbb{N}|$ and $\mu(A) = \infty$ if A is not countable on the sigma ...
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Borel Sigma-Algebra on $\mathbb{R}$

Show that the Borel sigma-algebra on $\mathbb{R}$, denoted $B_R$ is generated the open intervals in $\mathbb{R}$. My attempt: Let $I$ be the collection of all open intervals, let $\sigma I$ be the ...
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Why is there a $A_{\varepsilon}\subseteq C$ such that $\nu(C\setminus A_{\varepsilon})<\varepsilon$?

Let $E$ be countably infinite. Suppose that $v$ is a probability measure on $E$ supported on $C\subseteq E$. Since $\nu(E)=\nu(C)=1$ for any $\varepsilon > 0$ there exists a ...
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Why does from $v(i)\leq u(i)$ and $u(E)=1$ follow that $u$ is unique?

Let $E$ be countably infinite and $v$ a probability measure on it with support $C\subseteq E$, i.e. $v(E)=v(C)=1$. Let $u$ be another measure on E with $u(E)\leq 1$. Now it was shown that $v(i)\leq ...
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Is the integral of a measurable function measurable (wrt. a parameter)

Let $u \in L^2(0,T;L^2(\Omega))$ on a compact Riemann manifold $\Omega$. Is it true that $$t \mapsto \int_\Omega u(t)w$$ is measurable for $w \in L^2(\Omega)$?
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Reasoning about absolute continuity between two measures by reasoning about their supports

Suppose we have two measures $\mu, \nu$ on $(\mathbb{R}, \mathcal{B}(\mathbb{R}))$ with $\text{supp}(\mu) \subseteq \text{supp}(\nu)$, where $\text{supp}(\mu)$ denotes the support of $\mu$, i.e. $$ ...
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40 views

Proving $\frac{X_n}{C_n}\rightarrow 0$ with probability 1

Suppose ${X_n}$ is a sequence of real valued random variables. Show that there is a sequence of constants ${C_n}$ s.t. $\frac{X_n}{C_n}\rightarrow 0$ with probability 1 I'm not entirely sure how ...
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Transformation theorem: calculate picture of a set

I have this function: $T:(0,\infty)^2 \rightarrow T((0, \infty)^2), \quad T(x,y)=\left( \frac{y^2}{x},\frac{x^2}{y} \right)$ Now I try to estimate $T(M)$ with: $0<p<q, \quad 0<a<b$ ...
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17 views

Integration with respect to a measure

I am trying to get an explanation in words, or math, of what the $d\mu$ means in an integration statement. Such as: $$\int f \ d\mu$$ How does the measure change our old "calculus" notion of ...