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

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32
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0answers
1k 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 ...
16
votes
0answers
308 views

We have sums, series and integrals. What's next?

We know how to sum or average a finite number of terms: sums. We know how to sum a countable infinite number ${\beth_0}$ of terms: series. We know how to sum ${\beth_1}$ terms: integrals. How to ...
16
votes
0answers
456 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 ...
14
votes
0answers
798 views

Errata for Dieudonné's Treatise on Analysis volume 2 second edition

I was looking again at the beautiful and quite complete work of Dieudonné, his Treatise of Analysis, to refresh my memory about some aspects of classical analysis. I especially love this Treatise for ...
11
votes
0answers
100 views

Given a space, is there a notion of “how many” open sets contain a given point?

Say $(X,\tau)$ is a topological space and pick some point $x\in X$. Define $U_x := \{U\in \tau \;|\; x \in U \}$. Is there a way to put a measure on $\tau$ so that we can meaningfully compare the ...
11
votes
0answers
997 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 ...
11
votes
0answers
323 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 ...
10
votes
0answers
342 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 ...
10
votes
0answers
393 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 \...
10
votes
0answers
777 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 ...
10
votes
0answers
925 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 ...
10
votes
0answers
311 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 ...
9
votes
0answers
104 views

Halmos Measure Thoery section 62 exercise 3

Is there a locally compact group $G$ and a Borel measure $\mu$ on $G$ such that \begin{equation*} H=\{g\in G\mid \mu(gE)=\mu(E) \: \text{for all measurable} \: E\} \end{equation*} is not a closed ...
9
votes
0answers
141 views

Lebesgue Premeasure via Transfinite Induction

If $I=[a,b)$ we write $|I|=b-a$ for the length of $I$. Given a theorem of Caratheodory, the tricky part in showing the existence of Lebesgue measure is this: Lemma If $[0,1)$ is the disjoint union of ...
9
votes
0answers
199 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 $\bigcup_{n=1}^\infty(q_n-2^{-...
9
votes
0answers
381 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 ...
9
votes
0answers
432 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 $\mathcal{D}(\mathbb{R}^n)$...
9
votes
0answers
564 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 $\...
9
votes
0answers
348 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|} (x)...
8
votes
0answers
118 views

Fast convergence in $L^1$ implies convergence almost everywhere

This is a proof-verification request. Claim: Let $(X,\mathscr M,\mu)$ be a measure space. Let $f_n$ ($n\in\mathbb N$) and $f$ be measurable, integrable, real-valued functions such that $(f_n)_{n\...
8
votes
0answers
119 views

Transformations of RV's Ensuring Absolute Continuity of Quantile Functions

Given a real random variable $X$, suppose $T:\mathbb{R}\to\mathbb{R}$ is non-decreasing. Define $Y=T\left(X\right)$. Let $Q_{X}$, $Q_{Y}$ be the corresponding right-continuous quantile functions. ...
8
votes
0answers
218 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. ...
8
votes
0answers
387 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
43 views

Maximal ideals in the ring of measurable functions

The $R$ ring of continuous functions from $[0,1]$ to $\mathbb{R}$ has a property that its maximal look like a subset of $R$ consisting of those functions which vanish at a common single point in $[0,...
7
votes
0answers
254 views

Relation between Shannon Entropy and Total Variation distance

Let $p_1(\cdot), p_2(\cdot)$ be two discrete distributions on $\mathbb{Z}.$ Total variation distance is defined as $d_{TV}(p_1,p_2)= \frac{1}{2} \displaystyle \sum_{k \in \mathbb{Z}}|p_1(k)-p_2(k)|$ ...
7
votes
0answers
104 views

Let $(X,\Sigma, \mu)$ be a measure space. If $f$ is integrable and $\int_E {f}\,{d\mu} = 0$ for all $E\in\Sigma$ then $f=0$ almost everywhere.

My attempt: I called: $A = \{ x\in X : f(x) \ne 0 \}$ $B_n = \{ x\in X: f(x) \gt \frac{1}{n} \}$ $C_n = \{ x\in X:f(x) \lt -\frac{1}{n} \}$ $B = \bigcup_{n=1}^{\infty}{B_n}$ $C = \bigcup_{n=1}^{\...
7
votes
0answers
287 views

Sigma-algebra generated by a set of random variables

I know from standard textbooks that "Given the measurable functions $X_i:(\Omega,\mathcal{F})\rightarrow(\Omega_i,\mathcal{A}_i)$, the $\sigma$-algebra generated by a set of random variables $(X_i; i\...
7
votes
0answers
122 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
251 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 E_k\...
7
votes
0answers
650 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 ...
6
votes
0answers
57 views

If the difference of two independent random variables has a mean, so does each variable

This is a proof-verification request; I’m also recording this proof for my own later reference. Any feedback is appreciated. Claim: Let $X$ and $Y$ be independent, real-valued random variables on ...
6
votes
0answers
75 views

Capacity vs measure of a set - intuitive understanding

There is a concept of measure of "largeness" of a set, called capacity. The intuition is, instead of physical largeness (measured by Hausdorff or Lebesgue measure), capacity measures how good a given ...
6
votes
0answers
64 views

Can a Banach measure be consistent with Friedman's Fubini-type theorem for non-measurable functions?

I read on Wikipedia that Harvey Friedman proved that the following is consistent with ZFC+¬CH: For all functions $f:[0,1]^2 \mapsto \mathbb{R}^+$ such that both $\int_0^1 \left ( \int_0^1 f(x,y) dy \...
6
votes
0answers
68 views

Theorem about the liminf and limsup

Its apparently referred to as Besicovich theorem (I checked all of his available papers but they only one that seems relevant is related to Hausdorf measure and the set E is in the plane) : for $E\...
6
votes
0answers
201 views

Approximating a discrete measure with a continuous one

In physics it is common to approximate distributions of point masses or charges with continuous distributions. To do this, one typically defines a density function by moving throughout the space a ...
6
votes
0answers
172 views

Anti-random reals

EDIT: This has now been crossposted at MO: http://mathoverflow.net/questions/219366/antirandom-reals. This is partially motivated by my question at mathoverflow: http://mathoverflow.net/questions/...
6
votes
0answers
150 views

Does $\pi$ contain infinitely many “zeros” in its decimal expansion?

Some number doesn't contain $"7"$ in its decimal expansion. For example Liouville's constant $$L=\sum_{n=1}^\infty\frac{1}{10^{n!}}=0.11000100....$$ contains only $0$ and $1$. It is well-known ...
6
votes
0answers
181 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
246 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: https://www.encyclopediaofmath....
6
votes
0answers
168 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 ...
6
votes
0answers
140 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
148 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
139 views

Linear w.r.t. any measure

Let $X$ be a Banach space endowed with a Borel $\sigma$-algebra. How do we call a real-valued Borel function $f$ that satisfies for any Borel probability measure $\mu$ the following formula $$ \...
6
votes
0answers
102 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
176 views

Baire sets of $X$ possess the required Cartesian product property

Let $X=X_{1}\times X_{2}$ is locally compact space, and define $$E=\{E_{1}\times E_{2}\;|\; E_{i}\; \text{is a Borel set in}\; X_{i}\; ,\; \text{for}\; i=1,2\}$$ Now why the Baire sets of $X$ are in ...
6
votes
0answers
59 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
113 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
441 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
135 views

Demystifying invariant measures in probability theory

Just trying to understand, at least conceptually, invariant measures, and specially their role in probability theory. To be brief: I understand that if we have a set $X,$ with $A \subset X$ just a ...
5
votes
0answers
61 views

Difference between $d\mu(x)$ and $\mu(dx)$

In my lecture notes of probability course I found two different notations involving $d,\mu$ and $x$: is there any difference between $\mu(dx)$ and $d\mu(x)$? For example I read $\mu(dx) = \frac{1}{\...