# Tagged Questions

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

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 ...
355 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 ...
759 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 ...
285 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 ...
769 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 ...
855 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 ...
313 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 ...
335 views

589 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 ...
59 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 ...
192 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 ...
81 views

139 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 $m(X)=1$. ...
144 views

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, $D_n = \{kp^{-n}\colon 0 \leq k < p^n ... 0answers 98 views ### Is$\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$such that ... 0answers 175 views ### Baire sets of$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 ... 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). ... 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 ... 0answers 431 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 ... 0answers 77 views ### Concavity of the$n$th root of the volume of$r$-neighborhoods of a set Let$A$be a closed subset of$\mathbb{R}^n$. For$r>0$, let$A_r$be the$r$-neighborhood of$A$, namely the set$\{x:\operatorname{dist}(x,A)\le r\}$. Is the function$f(r) = \mu(A_r)^{1/n}$... 0answers 54 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 ...
I am in the midst of proving the following Theorem Let $(f_n)$ be a sequence of measurable functions converging almost everywhere to a function $f$ on a measurable set $E\subset [0,1]=\Omega$. Then ...