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

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Measure Algebra Question

Background: Let $G$ be a locally compact group, and define $M(G)$ to be the vector space of all regular complex measures defined on $G$, normed by total variation. For $\mu,\nu\in M(G)$, define the ...
2
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1answer
56 views

Find a measurable function such that $f(x)\le \alpha$ for $x\in E_\alpha$

Theorem: Given $\{ E_\alpha \}_{\alpha \in \mathbb{R}}\subset \mathcal{M}$ such that $E_\alpha \subset E_\beta$ for $\alpha < \beta$. We have also that $\bigcup_{\alpha \in \mathbb{R}}E_\alpha=X$ ...
2
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1answer
165 views

Conditional Expectation with independent sub-sigma fields

Let X and Y be bounded random variables on a probability space $(\Omega, \mathcal{F}, \mathbb{P})$. Consider two independent sub-$\sigma$ fields $\mathcal{G}$ and $\mathcal{H}$ of $\mathcal{F}$. We ...
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2answers
250 views

Rigorous proof that $\int_{\Omega}X\;dP=\int_{-\infty}^{\infty}xf(x)\;dx$

I'm trying to prove rigorously that $\int_{\Omega}X\;dP=\int_{-\infty}^{\infty}xf(x)\;dx$. Where $f$ is the pdf of the random variable $X$. I can't find a proof on the wikipedia article, or if it's ...
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1answer
365 views

Can the Fubini theorem be applied to a trapezoid?

I have two questions: How does one apply the Fubini theorem? and Can it be applied to a trapezoid? Thank you.
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2answers
76 views

Lebesgue measure nullsets

I have a question concerning nullsets of the Lebesgue measure $\lambda^d$ on $\mathbb{R}^d$. I think the set $$\{(x_1,\ldots,x_n)\in\mathbb{R}^n: \sum_{i=1}^n x_i=0\}$$ is a nullset of the ...
1
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1answer
118 views

A generalization of the conditional expectation to kernels

Let $\left(\Omega_1,\mathcal{A}_1, P\right)$ be a probability space, let $\left(\Omega_2,\mathcal{A}_2\right)$ be a measurable space and let ...
3
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1answer
481 views

An alternate proof of Egorov's Theorem

I came up with a proof of Egorov's Theorem which I can't find in my books or on the net, which makes me think it's wrong, which means I have a misunderstanding somewhere which needs to be hammered ...
3
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0answers
175 views

Calculating a simple integral using abstract measure theory

Let $(X,\mu)$ be a measure space. For a measurable function $\iota:X\to X$, we have for all measurable $E\subset X$, $$\int_{\iota^{-1}(E)}g\circ\iota \, d\mu = \int_{E}g \, d(\iota^{*}\mu),$$ where ...
1
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1answer
530 views

$\liminf, \limsup$, Measure Theory, show: $\lim \int n \ln(1+(f/n)^{1/2})\mathrm{d}\mu=\infty$

Let $(X,\Omega,\mu)$ be a measure space and $M^+(X,\Omega)$ denote the set of all non-negative real valued measurable functions. If $f \in M^+(X,\Omega)$ and $0< \int f \mathrm{d}\mu < ...
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1answer
585 views

Definition of a dominating function and the Dominated Convergence Theorem.

I apologise if this is a rather simplistic or even silly question, but I am confused with the word "dominated" in Lebesgue's Dominated Convergence Theorem (DCT) since I can find no definition of a ...
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1answer
174 views

Proof that $dx/|x|$ is a Haar measure on non-zero reals?

Most importantly, what is the meaning of this notation $\lambda = dx/|x|$? How do I compute say $\lambda(0,1)$ for example?
4
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1answer
53 views

Existence of variation

Let $I[w] =\int_U L(Dw,w,x) dx$. Let $1<q<\infty$, and there exist constants $\alpha>0$,$\beta\ge0$ such that $$L(p,z,x)\ge \alpha |p|^q - \beta$$ This implies that if $I[w]$ exists, $$I[w] ...
3
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1answer
203 views

Measurable function of a transformation still measurable

I'm looking for a maybe simpler or more elemental proof of the following statement: Let $f:\mathbb{R^n}\to \mathbb{R}$ be (Lebesgue-)measurable. Then $F:\mathbb{R^{2n}}\to \mathbb{R}$ such that ...
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2answers
72 views

Approximation for $L_{\text{loc}}^{\infty}(U)$ is this proof correct?

Let $U\subset\mathbb{R}^n$ be open and bounded. I am trying to extend Evans' proof (in his PDE book) for approximating functions in $L_{\text{loc}}^{p}(U)$ for the case that $p=\infty$ using ...
2
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1answer
81 views

Tails of family of integrable functions

It is well known that tail of an integrable function on $\mathbb{R}^d$ is small, i.e., Given $\epsilon>0$, there is $R>0$ such that $$\int_{\{|x|>R\}}|f(x)|dx<\epsilon.$$ I was wondering ...
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1answer
96 views

A theorem about the Poisson Point process.

In the proof of the Levy-Khintchine theorem, I saw a theorem about the Poisson point process. The theorem states that if $\Pi$ is a poission point process on $S$ with intensity measure $\mu.$ Let ...
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1answer
254 views

Non-ergodic measure

Is there an easy way to see that if $\mu$ and $\nu$ are $T$-invariant measures on the same space $X$, and $\mu \neq \nu$, then $\frac{1}{2}(\mu+\nu)$ is NOT ergodic? I know that ergodic measures are ...
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1answer
44 views

construct a lebesgue integral function

Construct a Lebesgue integrable function $f$ with the property that for any interval $I$ in $\mathbb{R}$ (real line) and any $M >0$, The measure $|{x \in I: |f(x)| > M}| > 0$ How would I go ...
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1answer
72 views

Is it true that $ \int_{[1,\infty)} f_n\to \int_{[1,\infty)} f$?

Can you please help me solve this on measure theory? My TA did not go over this. He said we are not going over this but you can do this if you want. Can someone please explain to me? Thanks. Suppose ...
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1answer
794 views

Stochastic process, Gaussian, with zero mean is a Wiener process

Let $(\Omega, \mathcal F , \mathbb P)$ be a probability space and let $\mathcal F = \{\mathcal F_t\}_{t\ge} $ a filtration. Let $W=\{W_t;t ≥ 0\}$ be a stochastic process adapted to $\mathcal F$. ...
2
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0answers
321 views

Euler's theorem for homogeneous functions

Let $\textbf{R}_{+}$ be the set of positive real numbers. The following is a well-known theorem due to Euler: A differentiable function $f:\textbf{R}^n_{+} \rightarrow \textbf{R}_{+}$ is positively ...
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1answer
255 views

Embedded Lp spaces [duplicate]

Let $L^\infty(Ω,F,P)$ be the vector space of bounded random variables $(X ∈ L^\infty (Ω,F,P)$ means that there exists a constant C such that $|X(ω)|≤C$, a.s.$)$. Show that ...
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1answer
65 views

$\int_\Omega F(u_n)\to0$ implies $\int_\Omega F(au_n)\to 0$ for $a\in [0,\infty)$?

Let $\Omega\subset\mathbb{R}^N$ be a bounded domain and $F:[0,\infty)\to [0,\infty)$ be a convex, strictly increasing and continuous function satisfying $F(0)=0$. Suppose that $u_n\in ...
2
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1answer
103 views

Does a product measure on a product space constructed from two sub-fields of the same space determine a measure on the underlying space?

Let $\mathcal{A}_1,\mathcal{A}_2$ be $\sigma$-algebras on $\Omega$. Let $P$ be a probability on $\mathcal{A}_1$ and let $Q$ be a Markov kernel from $\mathcal{A}_1$ to $\mathcal{A}_2$. Set $K:=P\otimes ...
5
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2answers
116 views

Evaluation of “concentration of measure function” at zero.

Let $(X,d)$ be a metric space equipped with a probability measure $\mu$ (defined on the Borel $\sigma$-algebra on the topology induced by the metric $d$). We define the concentration function of the ...
4
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2answers
624 views

Metric assuming the value infinity

If we instead define a metric as $d:X\times X \rightarrow [0,\infty]$, do we lose any nice properties of metric spaces? The reason I ask is that I saw this theorem: Given a finite measure space ...
1
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1answer
102 views

Show that a function is continuous

Let K be bounded and continuous and bounded on $\mathbb{R}^{n}$ and let $f$ be Lebesgue integrable on $\mathbb{R}^{n}$. Show that the function $g$ defined on $\mathbb{R}$ by $g(t) = ...
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1answer
42 views

Can't establish a lower bound on a supremum

I have a sequence of functions $f_{k,j}:[0,1]\to\mathbb{R}$ defined by $$f_{k,j} = k^{\frac{1}{p}}\chi_{(\frac{j-1}{k},\frac{j}{k})},$$ for all $k\geq 1,1\leq j\leq k$. This serves as an example of ...
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1answer
79 views

Measuring distances on any coordinate system

I was reading the book The ABC of Relativity from Betrand Russell, and at some point, the author mentions a method for measuring the distance between 2 points on any coordinate system. He says that ...
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2answers
205 views

Riemann integral and Lebesgue integral

$f:R\rightarrow [0,\infty)$ is a Lebesgue-integrable function. Show that $$ \int_R f \ d m=\int_0^\infty m(\{f\geq t\})\ dt $$ where $m$ is Lebesgue measure. I know the question may be a little dump. ...
3
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1answer
120 views

Egorov's Theorem for $\mathbb{R}^n$

Let $g,f,f_n:\mathbb{R}^n\to \mathbb{R}$ and $\{f_n\}_{n=1}^\infty$ be a sequence of functions that have the following property: i) $f_n \to f$ a.e ii)$|f_n(x)|\leq g(x)$ a.e and $g \in ...
3
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1answer
120 views

Weak $L^{p}$ spaces are quasi-normed?

Let $(X,\mathcal{B}, \mu)$ be a measure space. Then for $0< p < \infty$ by definition $L^{p,\infty}(X,\mathcal{B}, \mu)$ is the class of all measureable functions $f$ such that ...
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2answers
1k views

Proof of Egoroff's Theorem

Let $\{f_n \}$ be a sequence of measurable functions, $f_n \to f$ $\mu$-a.e. on a measurable set $E$, $\mu(E) < \infty$. Let $\epsilon>0$ be given. Then $\forall \space n \in \mathbb{N} \space ...
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1answer
279 views

Why is ess sup $f$ not ess max $f$?

Consider a measure space $(X,\Sigma\,\mu)$. Given that one can easily prove that, $\mu$-a.e., $f \leq \text{ess} \sup_X f$, why is the notation not simply "$ \text{ess} \max_X f$"? (Here $\text{ess} ...
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1answer
113 views

How many Borel conjectures are there

The following may be referred to as Borel conjecture: Every strong measure zero set of reals is countable. On the other hand Wikipedia refers to the following as the Borel conjecture: Let $M$ and ...
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2answers
199 views

Definition of Lebesgue-Stieltjes measure on $\mathbb R$

Let $F:\mathbb R\to\mathbb R$ be a non-decreasing, left-continuous function. Let $a,b\in\mathbb R$, then define the Lebesgue-Stieltjes measure $$ m[a,b]=F(b+)-F(a), \quad m(a,b)=F(b)-F(a+) $$ ...
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1answer
161 views

Proving that for $f\geq0$ on $X$, $\int_X f d\mu = 0$ iff $f = 0$ a.e.

Okay, so the question is the following: Suppose $f \geq 0$ is a measurable function on the measure space $(X,\Sigma,\mu)$. Prove that \begin{align} \int_X f d\mu = 0 \text{ if and only if } f = 0 ...
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0answers
161 views

lebesgue measure of difference of two sets [duplicate]

Suppose $A,B$ are in $\mathbb R^n$, $A$ lebesgue measurable and $|(A \setminus B)\cup(B\setminus A)| = 0$. Show $B$ must be lebesgue measurable as well and $|A| = |B|$ (where $|\cdot |$ is the ...
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1answer
105 views

measure of limsup positive for infinitely many terms

let $(X, M, \mu)$ be a measure space, and let ${E_k}$ be in $M$. if $\mu(\cup E_k) < \infty$, and $\mu(E_k) \ge c > 0$ for infinitely many $k$'s for some $c$, then $\mu(\limsup E_k) > 0$. ...
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1answer
61 views

$g(x) = 0$ if $f(x)$ is rational measurability

Say $f$ is a measurable real-valued function on $X$, and put $g(x) = 0$ if $f(x)$ is rational and $g(x)=1$ if $f(x)$ is irrational. Is $g$ measurable?
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1answer
452 views

Composition of Lebesgue measurable function $f$, with a continuous function $g$ having a certain property, is Lebesgue measurable

Suppose that $f$ is Lebesgue measurable and $g$ is real valued, continuous, and has the property that for any null set $N$, $g^{-1} (N)$ is measurable. Then $f \circ g$ is also Lebesgue measurable. ...
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1answer
581 views

Need an explanation of this paragraph “Measure Theory”

I will just quote a part of one proof in "On uniformly regular topological measure spaces by Babiker: page 781" vol43 No4 Duke Math. J. 1976. Let $I$ be the unit interval endowed with Lebesgue ...
2
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1answer
76 views

Uniformly regular measure “Babiker”

A regular Borel (Radon) probability measures $\mu$ on compact Hausdorff space $X$ is called uniformly regular if: There is a countable family $\mathcal{A}$ of compact $G_\delta$-subsets of $X$ ...
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1answer
55 views

Is this function measurable? Something to do with Bochner space and norms.

Suppose $f:[0,T]\to X$ is a measurable map where $X$ is Hilbert space. Suppose also that $R(t):X \to X^*$ is an isometric isomorphism with $$\lVert R(t)f(t)\rVert_{X^*} = \lVert f(t) \rVert_X$$ also ...
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1answer
75 views

A question about “nice” functions

Let $f:\mathbb R \rightarrow \mathbb R$ be a function such that $\lambda(I)=\lambda(f(I))$ for each interval $I \subseteq \mathbb R$. ($\lambda$ is Lebesgue measure here.) Let us call such functions ...
2
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0answers
147 views

Area of a Random Polygon

The following is a long description of a computation I'd like to make. You can think of the process described as a spider randomly building a web. I'd like to know how big we can expect the web to ...
2
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1answer
187 views

Integral over null set is zero but integral of Dirac delta function is 1

We know integral of any function over a null set is zero. But for Dirac delta function ($\delta=+\infty$ iff $x=0$ otherwise $\delta=0$) $$ \int_{-\infty}^{+\infty}\delta =\int_0^0\delta =1. $$ Is it ...
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2answers
72 views

If $\lVert f(t) \rVert:[0,T] \to \mathbb{R}$ is measurable, is $f$ measurable?

Let $f:[0,T] \to X$ be a mapping to a Banach space $X$. If its norm $\lVert f(t) \rVert$ is measurable, is $f$ itself measurable? The converse is true.
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1answer
103 views

Exercise on Lebesgue measure ( Treatise of Analysis Vol2 by Dieudonné)

Someone challenge me to bring the solution from anywhere! So I have posted here and see, I am optimist because this website is excellent and its members are so helpful. Let me start with this ...