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

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0
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1answer
14 views

Continuity at $x$ of increasing function if certain sequences exist

I'm working through the first few chapters of Royden-Fitzpatrick to learn measure theory and I got stuck on this question. Let $f$ be increasing on $I$, an open interval. Then for $x \in I$, $f$ is ...
0
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1answer
24 views

Square of absolute value of a function different than square of function

How come if f is measurable, we might have $|f|^2\neq f^2$? Can you provide an example? I think it is true if f is real.
0
votes
0answers
33 views

Girsanov's theorem and simulation of bond prices

Assume that we want to calculate the time $t=0$ price of a bond: $B(0,T) = E_P[\exp(-\int_0^T r_s ds)]$, where $r$ is the interest rate following the SDE $dr_t=k(\theta-r_t)dt+\sigma ...
3
votes
1answer
33 views

Lebesgue integral over “bad” measurable sets

Let $\Omega \subset \mathbb{R}^n$ ($n \geq 1$) be a bounded open domain and $f \in L^\infty(\Omega)$ possibly changes the sign. Assume that the set $$ \Omega^+ := \{x \in \Omega: f(x) > 0 \} $$ has ...
3
votes
1answer
21 views

bounded measure and dense subset of continuous functions

Let $C_0(R^n)$ be the space of continuous functions from $R^n$ to $R$ which vanish at infinity. Let $D$ be a subset of $C_0(R^n)$, I'd like to prove that if D is not dense in $C_0(R^n)$, then there ...
0
votes
2answers
38 views

Motivation behind the proeprties of sigma algebra

What is the motivation behind the class $B$ of all measurable sets to satisfy the following properties : 1) $A_1, A_2 \in B$ implies $A_1 \cup A_2 \in B$ 2) $\{A_n\} \in B $ and $\{A_n\}$ is ...
0
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1answer
26 views

Meaure Zero of the XY Plane

Prove that the xy-plane has measure zero in R3. I am learning about measure zero in Analysis and I understand that it would have measure zero if it can be broken down into a covering of rectangles. I ...
2
votes
4answers
144 views

every subset of a measurable set is measurable

Is it true that every subset of a measurable set is measurable? for any measure. So if A is a measurable set then, B as a subset of A must be measurable wrt the same measure.
1
vote
1answer
41 views

Two measures on a same space

I have two measure space $(X, S, \mu_1)$ and $(X, S,\mu_2)$, where $S$ is the minimal $\sigma$-algebra containing sets $T = \{E_i\}_{i \in I}$. Suppose further that $T$ is closed under taking finite ...
2
votes
1answer
54 views

Application of Riesz representation theorem

Suppose the following situation. We have linear functional $l$ on the space $H(\mathbb{C}^n)$ of entire function and wish to find a representation for $l$ with integration against a complex Borel ...
1
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0answers
59 views

The Cantor set is nowhere dense

I am considering the so called Cantor ternary set $C$ on $[0,1]$. I have just proved that its Lebesgue measure is $0$. To show that $C$ is nowhere dense, is it correct the following reasoning? By ...
4
votes
2answers
114 views

Intuition behind the failure of unimodularity

If $G$ is a locally compact group then up to normalization it admits a unique Haar measure: a left invariant measure defined on all Borel subsets of $G$, which assigns every compact set a finite ...
0
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0answers
14 views

iterated integrals using fubini's theorem

In the following questions $\mu$ is a product measure $\mu1 $ * $\mu2$ on ($\Omega$;$\mathscr{A}$) = ($\Omega$1*$\Omega$2;$\mathscr{A1}$ * $\mathscr{A2}$). is the counting measure on N (so that if ...
0
votes
1answer
19 views

Notation, abbreviation $a.s.$ measure theory

Could you tell me what $m - a. s.$ means in measure theory? Here $m$ is a measure. Thank you.
2
votes
1answer
16 views

Set of simple predictable processes is a vector space

I have a question, which is probably very easy for you to answer. How can I show that the set of simple predictable processes a vector space is? It's clear that I only have to show that the sum of ...
1
vote
0answers
41 views

Distribution function and decreasing rearrangement

Let $(X,dx)$ a measure space and $f\in L^p(X,\mathbb{C})$; let's define its distribution function $$F(\alpha)=meas(\{x\in X||f(x)|>\alpha\})$$ and the decreasing rearrangement ...
2
votes
1answer
40 views

measure on non-oriented Riemannian manifold

Let $M$ be a non-oriented Riemannian manifold of dimension $m$. Nash embedding theorem implies that there exists an isometric embedding $\phi: M\longrightarrow \mathbb{R}^n$ for $n$ sufficiently ...
0
votes
1answer
29 views

Borel-set, open, measurable function.

I have a questions about Borel sets. Here is how they defined in my book: Now they say that, the set consits of open sets. But it must not nececarrily be all open sets on X? The reason this ...
1
vote
2answers
54 views

Must every probability distribution over a countable set be discrete?

Intuitively I expect this to follow from countable additivity, but there are ideas I can't rule out such as: Select a real number r from the uniform distribution over [0, 1]. If r is exactly 0.5, ...
0
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0answers
17 views

product of measurable functions

in my course i saw that if $f$ and $g$ are both $B$-mesurable then $fg$ is also $B$ mesurable. but if $f$ is $B_{1}$-mesurable and $g$ is $B_{2}$-mesurable , $fg$ is measurable for what ...
1
vote
0answers
24 views

Differentiation of Radon measures

Assume $\ (X,d)$ is a locally compact metric metric space and $\ \nu,\, \mu$ are Radon measures on $X$. Then, suppose that the following hypothesis hold: $\ w\in ...
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1answer
36 views

Lebesgue Measurability of the Square of a Null Set [closed]

Suppose $A$ is Lebesgue measurable such that $\lambda(A)=0$, show that the set $B = \{x^2 | x \in A\}$ is also of Lebesgue measure zero.
1
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1answer
39 views

Probable use of Radon Measure

The problem is: Suppose $\mu$ is a positive Borel Measure on $\mathbb R^{1}$ which is finite on bounded sets. If $ \forall f,g \in C_{c} ( \mathbb R^{1})$ ; $ \int_{ \mathbb R^{1}} fg d\mu = (\int_{ ...
0
votes
0answers
21 views

Superadditivty of the Inner Measure

Define a set function $\mu^*$ by: $\mu^*(E) = \sup \{ \text{length}(F) | \text{ open interval }F \subseteq E \}$. My question is, is this set function subadditive? superadditive? Note that the ...
0
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1answer
29 views

Projection of a Manifold has measure zero

Let $M$ be a $k$-dimensional sub manifold of $\mathbb{R}^N$, and let $\pi_n:\mathbb{R}^N\to\mathbb{R}^n$ be the canonical projection, with $n>k$. Can we show that $\pi_n(M)\subset \mathbb{R}^n$ has ...
4
votes
0answers
64 views

Is there a measure on $\mathbb{R^3}$ other than volume?

Of course there is the trivial measure where each subset of $\mathbb{R^3}$ have measure zero. But I am asking for a measure $\mu$ for which $\mu(E)=1$ where $E=[0,1]^3$. I thought about the radius, ...
0
votes
0answers
29 views

Infinitesimal generator of Markov jump process and delta measure

How can I prove that the following integral is null? \begin{equation} \int_{\mathbf{R}^{d}} \gamma\left(z\right) \delta_{z}\left(y\right), \qquad \gamma>0 \end{equation} I want to use this ...
1
vote
0answers
21 views

Probability space associated with a compact group

Is the probability space associated with a compact group with Haar probability always a standard probability space? I recall seeing somewhere the fact that if the topology generating the Borel sigma ...
4
votes
2answers
81 views

Borel sets and measurability

Is it always possible to construct a measure $ \mu $ on a Hausdorff space Y such that the $ \mu $-measurable sets are exactly the Borel sets of Y? By Theorem in 2.2.13 of Federer's book this question ...
2
votes
1answer
26 views

Finding σ-Algebra

Let $\Omega=[0,1]$ and $Y(w)=\begin{cases}1, & \text{if $w\in [0,1/3]$} \\2, & \text{if $w\in (1/3,1]$} \end{cases}$ What is the $\sigma$-Algebra created by $Y$, $\sigma(Y)$? I am kinda ...
1
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0answers
36 views

The support of nonnegative integer valued measure

Let $(E,\mathcal{E},\mu)$, where $E$ is a metric space, $\mathcal{E}$ is the Borel $\sigma$-algebra of $E$, and $\mu$ is a $\sigma$-finite nonnegative integer valued measure, i.e. $\mu$ is a ...
0
votes
1answer
43 views

Caratheodory's theorem and outer measure

I'm trying to show that $$\lambda(A)=\lambda(A\cap E)+\lambda(A\cap E^c)$$ where $\lambda$ is an outer measure, $A\subset \mathbb{R}$, $E \subset \mathbb{R}$, and $E$ is an elementary set; that is, ...
10
votes
0answers
257 views

Is this $really$ a categorical approach to $integration$?

Here's an article by Reinhard Börger I found recently whose title and content, prima facie, seem quite exciting to me, given my misadventures lately (like this and this); it's called, "A Categorical ...
0
votes
0answers
31 views

Existence of a Borel set with positive measure less than a fraction of the measure of each interval

Related to this. If $\mu$ denotes Lebesgue measure, is there a Borel set $A \subset \mathbb{R}$ such that $\exists\ \theta: 0<\theta <1$ $$0 < \mu(A \cap I) < \theta\cdot \mu(I)$$ for ...
1
vote
1answer
30 views

No Tonelli&Fubini contradiction

I was trying to solve the following question: Let $f(x,y)=\cases{1/x^2: x>y\ge0\\ -1/y^2: y>x\ge0\\ 0: x=y}$ Show that $\intop_0^1dx\intop^1_0f(x,y)dy\ne\intop_0^1dy\intop_0^1f(x,y)dx$ I did ...
2
votes
1answer
74 views

$f(z):=\int_{\mathbb{R}} \frac{1}{t-z} d\mu(t)$ show $\lim_{y\rightarrow 0}iyf(iy)=-\mu(\lbrace 0 \rbrace)$

I have some trouble with part b) Let $\mu$ be a finite Borel measure (i.e finite measure on the $\sigma$-algebra of Borel sets on $\mathbb{R}$). Define the function $$f(z):=\int_{\mathbb{R}} ...
0
votes
1answer
61 views

Prove a set contains an interval centered at zero.

Prove that if $E \subset [0,1]$ has positive measure, then the set $E-E = \{x-y : x,y \in E\}$ contains an interval centered around zero. Hint: consider the function $h(x)=\textbf{1}_{-E} \star ...
-1
votes
2answers
36 views

Lebesgue integral over Infinite measure sets

I would apreciate if someone could tell me wether this is true or false, or any advice on how to prove it or disprove it: Let $f$ be a positive measurable function over $(X,S)$ where S is a ...
2
votes
1answer
65 views

Can a countably generated $\sigma$-algebra be “approximated” by a $\sigma$-algebra generated by a countable partition?

My question is a bit vague, hopefully someone can still clarify. Let $(\Omega,\mathcal F,\mathbb P)$ be a probability space and assume that $\mathcal F$ is countably generated. My question is, does ...
1
vote
1answer
58 views

Application of Riesz representation theorem and norm of linear functional.

I think the solution to this question somehow involves Riesz Representation Theorem, but I don't see how to apply it. Suppose $\{X,\mathcal{M},\mu\}$ is a $\sigma-$ finite measure space, $1\leq ...
0
votes
1answer
21 views

integral convergence of function $e^{inx} f(x)$

Show that $n\to \infty \int_0^1e^{inx} f(x)dx=0$ for any continuous function on [0,1]. I tried to show the collection of {$e^{nix}:n\ge0$ } is algebra, separates points and do not vanishes at origin ...
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votes
2answers
50 views

matrix operator is bounded on $\ell^2$

Can anyone help me with this problem? Let $[a_{ij}]_{i,j=1}^{\infty}$ be an infinite matrix of real numbers and suppose that, for any $x \in \ell^2$, the sequence $Ax$ belongs to $\ell^2$. Prove ...
0
votes
1answer
17 views

A series aproximation for measurable functions

I've been working on this problem and I don't know how to face it. If $(X,M)$ is a measurable space and $f:\mathbb{X}\longrightarrow [0,\infty]$ is a measurable function then exist a collection of ...
0
votes
0answers
17 views

Sum of measures

Let $\alpha,\beta$ be measures on some $\sigma$-algebra $\mathcal A$ on a set $X$. I have to prove that $\mathcal L[\alpha+\beta] = \mathcal L[\alpha] \cap \mathcal L[\beta]$. I have already proven ...
2
votes
0answers
27 views

Measurability of a particular map

Let $X$ and $Y$ be standard Borel spaces and let $\mathcal P(X)$ denote the space of probability measures over $X$ endowed with the topology of weak convergence. Consider a map $f:X\times \mathcal ...
3
votes
1answer
30 views

A Problem in Convergence of Sequences of Random Variables

Let $\left( X_n \right)$ be a sequence of independent random variables on the measure space $(\Omega, \xi,\mathbb{P})$ with $$ \mathbb{P} \left( X_n=1 \right)= p_n \text{ and } \ \mathbb{P} \left( ...
1
vote
1answer
16 views

Hausdorff dimension and accumulation points on a smooth curve

I am wondering about the following, possibly naive, question. Suppose I have a smooth curve, which intersects the horizontal axis in a manner that leads to an accumulation point. More precisely, ...
9
votes
0answers
126 views

Topology of convergence in measure

Currently I am doing some measure theory(on $X=[0,1]$ with the Borel-Sigma algebra and the Lebesgue measure) and I am looking at sets$A \subset L^p$, such that for all $q \in (0,p)$, the topologies ...
0
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1answer
24 views

If for a.e $t \in (0,T)$, $u_n(t,x) \to u(t,x)$ for a.e. $x \in \Omega$,is $u_n \to u$ a.e. in $(0,T)\times\Omega$?

If for almost all $t\in (0,T)$, we have $$u_n(t,x) \to u(t,x) \quad\text{a.e. $x \in \Omega$}$$ does this mean that $$u_n \to u \quad\text{a.e. in $(0,T)\times\Omega$}?$$ Here $\Omega$ is an open ...
0
votes
1answer
19 views

Question about an outer measure of a real subset

Let B be a subset of the real numbers, and let $$\mu$$ be an outer measure, and let A be the union of a finite number of real intervals. I have to show that $$\mu(B)=\mu(B\cup A)+\mu(B\cap A^c)$$ ...