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

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17
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0answers
767 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 ...
11
votes
0answers
232 views

What properties are preserved under a measurable mapping?

Although in an abstract category the morphisms are not explicitly defined, in a concrete example (model theory?), morphisms are (always/usually?) mappings that preserve some properties. In the ...
10
votes
0answers
586 views

Compact set of probability measures

I think I can solve the following exercise if X is assumed to be separable, otherwise I can't. Let $X$ be a (Hausdorff) locally compact space, $\pi\colon X \to Y$ a continuous map into a ...
9
votes
0answers
260 views

Uniqueness of Brownian motion

May be it is a dumb question, but it vexed me a little bit. I understand the construction of the Brownian motion (first use Kolmogorov extension theorem to construct value at dyadic times and then use ...
9
votes
0answers
236 views

Restrictions of null/meager ideal

Let I denote the null/meager ideal on reals. Is it consistent that for any pair of non null/meager sets A and B, there is a null/meager preserving bijection between A and B? In particular, is this ...
9
votes
0answers
209 views

Measure-driven differential equations

Background: I need some help to understand the concept behind measure-driven differential equations. The solution of an ordinary differential equation is continuous. In order to describe discontinuous ...
9
votes
0answers
693 views

Cauchy-Formula for Repeated Lebesgue-Integration

Recently, I came across the following statements. They were annotated as consequences of Fubini's Theorem but neither proof nor reference were given. Let $f:[a,b]\times [a,b]\to\mathbb{R}$ be ...
9
votes
0answers
294 views

A question connected with the decomposition of a functional on $C(X)$ on Riesz and Banach functionals

Let $X$ be a metric space and let $C(X)$ be a family of all bounded and continuous functions from $X$ in $\mathbb{R}$. We call a positive linear functional $\varphi: C(X) \rightarrow \mathbb{R}$ the ...
8
votes
0answers
143 views

Explicit description of small open set containing the rationals

We know that the set $\mathbb{Q}$ of rational numbers has measure zero because it is countable. In fact, if $(q_n)_{n=1,2,\ldots}$ is an enumeration of $\mathbb{Q}$, then ...
8
votes
0answers
295 views

About devil's staircases

We say that a function $f:\left[a,b\right] \to \mathbb{R}$ is a singular function or a devil's staircase if $f$ satisfies the following properties: $f$ is continuous; $f(a) < f(b)$; $f$ is ...
8
votes
0answers
1k views

Monotone class theorem

I have some question about the Monotone Class Theorem and its application. First I state the Theorem: Let $\mathcal{M}:=\{f_\alpha; \alpha \in J\}$ be a set of bounded functions, such that ...
7
votes
0answers
244 views

Open problems in Federer's Geometric Measure Theory

I wanted to know if the problems mentionned in this book are solved. More specifically, at some places, the author says that he doesn't know the answer, for example :"I do not know whether this ...
7
votes
0answers
119 views

Uncountable family of random variables

Let $\{ \xi _a \}_{a \in [0;1]}$ be a family of independent uniformly distributed on $[0;1]$ random variables on some probability space $(\Omega, \mathscr{F},P)$, indexed by a continuous parameter. ...
7
votes
0answers
97 views

Connectedness of parts used in the Banach–Tarski paradox

A quote from the Wikipedia article "Axiom of choice": One example is the Banach–Tarski paradox which says that it is possible to decompose the 3-dimensional solid unit ball into finitely many ...
7
votes
0answers
216 views

Does this random variable have a density?

I have a persistent problem, which I'm almost certain can be answered using elementary probabilistic arguments, but for some reason I've been stuck for some time. Here is the problem. Let $(B_s, s ...
7
votes
0answers
296 views

Why is Lebesgue-Stieltjes a generalization of Riemann-Stieltjes? Moreover, is there an example where Lebesgue-Stieltjes is useful

I certainly have a question, but i don't know what the best title should be. Please edit the title if there is a better one :) And I believe, to get a better answer, it would be good to explain ...
7
votes
0answers
330 views

Egorov's theorem for this Lebesgue integral

I want to prove Egorov's theorem using this Lebesgue integral defined by the upper integral $$\int^*f:=\left\{\int h ; h \ge f \text{ and h upper-continuous }\right\}$$ $$\int_*f:=\left\{\int h ; h ...
7
votes
0answers
181 views

Carlson's model and Sierpinski sets

Suppose that every countably generated sigma algebra extending the Borel sigma algebra on $[0, 1]$ admits a measure extending the Lebesgue measure (This is true in Carlson's model which is obtained by ...
7
votes
0answers
1k views

Are vague convergence and weak convergence of measures both weak* convergence?

For quite a long time, I have been confused about the definitions of weak convergence and vague convergence of measures among other modes of convergence that root from functional analysis, mainly due ...
7
votes
0answers
224 views

Convergence of indicator functions in $L^2[0,1]$ when $m(\limsup(E_n)\setminus \liminf(E_n)) = 0$

I am trying to solve a qualifying exam problem. I would like to know what can be said about the convergence of the indicator functions $I_{E_k}$ in $L^2[0, 1]$ when it's known that $m(\limsup ...
7
votes
0answers
487 views

Equality condition in Minkowski's inequality for $L^{\infty}$

I am trying to find out when equality holds in Minkowski's inequality for $L^{\infty}$ (i.e. a necessary and sufficient condition for equality). I did a search and there was a discussion for the case ...
7
votes
0answers
299 views

Why is the Radon-Nikodym derivative needed for products of complex measures?

Given $\mu, \nu$ complex Borel measures on $\mathbb{R}^n$, then product measure $\mu \times \nu$ on $\mathbb{R}^n \times \mathbb{R}^n$ is defined by $$d(\mu \times \nu)(x, y) = \frac{d\mu}{d|\mu|} ...
6
votes
0answers
145 views

Lebesgue Differentiation Type Result

Let $\mathcal{B}$ be a collection of bounded, measurable subsets of $\mathbb{R}^{n}$, such that for every $x$ there exists a sequence $\left\{R_{k}(x)\right\}\subset\mathcal{B}$ containing $x$ and ...
6
votes
0answers
87 views

Why does the union of all open null sets is itself a nullset for second countable space?

On the online Encyclopedia of mathematics, it is written "The existence of a countable base guarantees that the union of all open μ-null sets is itself a nullset." See: ...
6
votes
0answers
88 views

Borel measurability is a local property

I am looking at Exercise 5.2 (page 44) in "Real Analysis for Graduate Students" by Richard Bass. Let $f:(0, 1)\to \mathbb{R}$ be a function such that for every $x\in (0, 1)$, there exist ...
6
votes
0answers
128 views

Finitely additive function on an infinite set, s.t., $m(A)=0$ for any finite set and $m(X)=1$ (constructive approach)

Other exercise which I found in Dudley's Analysis book: Show that there is a measure on a infinite set $X$, defined on $2^X$ s.t. is finitely additive, $m(A)=0$ for any finite set and $m(X)=1$. ...
6
votes
0answers
130 views

Question about B. Host paper 'Nombres, normaux entropie, translations'

I was reading this paper and I got stuck in a detail left for the reader that I couldn't figure out: Let $X = \mathbb{R}/\mathbb{Z}$, $p > 1$ a integer, $D_n = \{kp^{-n}\colon 0 \leq k < p^n ...
6
votes
0answers
88 views

Is $\sigma$-finiteness really a necessary condition for this problem?

Question: Let $(X, \mathcal A, \mu)$ be a measure space and suppose $\mu$ is $\sigma$-finite. Suppose $f$ is integrable. Prove that given any $\varepsilon$, there exists a $\delta >0$ such that ...
6
votes
0answers
125 views

Theorem $2.14$ page $40, 41$ in Rudin - Real and Complex Analysis

Can anyone tell me the signification of Theorem $2.14$ (The Riesz Representation Theorem in locally compact Hausdorff spaces), page $40, 41$ in Rudin - Real and Complex Analysis? And some applications ...
6
votes
0answers
63 views

Radon-Nikodým (write the density as a limit)

Let $\mu$ be a probability measure and $\nu$ a $\sigma$-finite measure on $(\mathbb{R},\mathcal{B})$ with $\nu\ll\mu$. Show that it is $\mu$-a.s. $$ \lim_{h\to 0}\frac{\nu [x-h,x+h)}{\mu ...
6
votes
0answers
632 views

Azuma's inequality to McDiarmid's inequality?

I was going through some notes on concentration inequalities when I noticed that there are two commonly-cited forms of McDiarmid's inequality. Long story short: I know how to prove the weaker one from ...
6
votes
0answers
198 views

Product of complex measures

Let $\lambda, \mu$ be complex measures on $(X,\alpha)$ and $(Y,\beta).$ Prove there exist a unique complex measure $\lambda\times \mu$ on the sigma algebra $\alpha\otimes \beta,$ such that ...
6
votes
0answers
458 views

Generated sigma algebra from Brownian Motion

Suppose that we have a Brownian motion and we define the P-augmented filtration by $$\mathcal{F}^W_t:=\sigma(\mathcal{F}^0_t \cup \mathcal{N})$$ where $\mathcal{F}_t^0:=\sigma(W_s;s\le t)$ and ...
6
votes
0answers
57 views

Shift operator on locally compact groups

Assume $f:G\rightarrow H$ is a measurable function between two locally compact abelian groups and let $T^h(f) = f\circ T^h$, where $T^h(x) = x-h$ (group operations in G and H are written additively). ...
6
votes
0answers
110 views

Haar measure existence using distributions

The book of Lieb and Loss proves the existence of Lebesgue measure in an unorthodox way as theorem 6.22, using the fact that ''positive distributions are measures''. My question is, whether it is ...
6
votes
0answers
371 views

Rudin's Complex and Real Analysis: Regular measure

This is for anyone who has Rudin's Real and Complex analysis book at hand. I was looking at Rudin's Theorem 2.17 and 2.18. So far everything makes sense, except for one statement that Rudin makes in ...
5
votes
0answers
68 views

Integral of a function over the Koch Curve. Is it rigourous enough?

(I want to investigate the validity of this approach, as I already know this is the correct result) I present a proof that $$\int_{K} (x+y) \ \mu(x,y)={{9+\sqrt 3} \over 18}$$ Where the region of ...
5
votes
0answers
77 views

The completion of the Borel $\sigma$-algebra the same as the completion of the Lebesgue outer measure?

My study group and I were discussing this question today. We can construct the Lebesgue measure using Caratheodory's extension theorem in the usual way: Given the function $F(x) = x$, we can ...
5
votes
0answers
53 views

$\sigma$-algebra $\mathcal{G} \subset \mathcal{F}$ is countably generated iff there is a random variable $X$ such that $\mathcal{G} = \sigma(X)$.

Where can I find a reference to the proof of the fact that a $\sigma$-algebra $\mathcal{G} \subset \mathcal{F}$ is countably generated if and only if there is a random variable $X$ such that ...
5
votes
0answers
79 views

$f$ is Riemann integrable iif the set of discontinuous points of $f$ has Lebesgue measure zero

This is a well known result in mathematics, but it's my first time attempting to prove it. I'm following the second book of Analysis from Folland. Below are the notations used and the theorem, from ...
5
votes
0answers
78 views

Standard machine in measure theory

Step 1.Prove the property for $h$ which is an indicator function. Step 2.Using linearity, extend the property to all simple positive functions. Step 3. Using Monotone property extend the ...
5
votes
0answers
89 views

The “muscle” behind the fact that ergodic measures are mutually singular

This is really motivated by the soft question at the end, but let me begin with something more circumscribed: Let $(X,\mathcal{B})$ be a measurable space and let $T:X\circlearrowleft$ be a self-map ...
5
votes
0answers
142 views

Why is the derivative of the translates of a measure measurable?

Let G be a topological group and X a measure space. Let $G \times X \rightarrow X$ be a measurable group action, $\mu$ a $\sigma$-finite measure on $X$, and $g\mu$ (for any $g \in G$) the measure ...
5
votes
0answers
74 views

An inequality between integrals of series of characteristic functions of cubes

Let $1\leq p<\infty$. Prove that there exists $C>0$ such that $$ \left(\int\left|\sum_{i=1}^\infty a_i\chi_{2Q_i}\right|^p \, dx\right)^{1/p} \leq C\left(\int\left|\sum_{i=1}^\infty ...
5
votes
0answers
112 views

Do most nowhere dense sets have measure $0$?

Inspired by this question here and in particular the answer I was wondering: Do most nowhere dense sets have measure zero? By "most" I mean in the sense of the "measure" of the set of all nowhere ...
5
votes
0answers
75 views

Looking for a a measure-theoretic treatment of “differential entropy”

If $X$ is a discrete random variable, its entropy $H(X)$ is usually defined as something along the lines of $-\sum \def\P{\mathbb{P}}\P(x) \log_2( \P(x))$, where the sum ranges over all the possible ...
5
votes
0answers
83 views

representation theorem on the path space

I'm working on a project and have done some work. However, there are some point where I'm unsure if my thoughts are correct. It would be appreciated if someone could share their thoughts about it. ...
5
votes
0answers
274 views

If a Radon measure is a tempered distribution, does it integrate all Schwartz functions?

The question might at first sight sound like the answer is trivially "yes", so let me clarify the question a bit. Consider given a Radon measure $\mu$ on $\mathbb{R}^n$. Let ...
5
votes
0answers
115 views

Krull dimension and Hausdorff dimension

Let $V$ be an irreducible algebraic variety over $\bf{C}$. Then the Krull dimension of the associated coordinate ring, its dimension as a topological space with the Zariski topology, and its Morley ...
5
votes
0answers
169 views

Don't understand this proof of equivalence of weak solutions to PDE

I'm trying to understand the proof that (c) implies (a) here in the following proposition (here, $\mathcal{V} = L^2(0,T;V)$). See the very last line in the image for that part: $$$$ $$$$ I give here ...