# Tagged Questions

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

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### 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 ...
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### About measures which respect half-spaces

I was wondering if there are measures known on $\mathbb{R}^n$ which somehow "nicely" see the half-spaces. I am not sure how to exactly quantify "nicely" and hence feel free to make your own ...
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### Characterization of measurable functions

Let $X$ be an uncountable set, let $\mathfrak{M}$ be the collection of all sets $E\subset X$ such that either $E$ or $E^c$ is at most countable, and define $\mu(E) = 0$ in the first case, $\mu(E) = 1$ ...
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### Computing $\int_0^{\infty}\frac{\cos(x)-1}{x^{1+\alpha}}\,\mathrm d x$

Let $\alpha\in(0,1)\cup(1,2)$. I want to show that the integral $$\int_0^{\infty}\frac{\cos(x)-1}{x^{1+\alpha}}\,\mathrm dx$$ exists (in Lebesgue’s sense: the integral of the absolute value of the ...
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### Can anyone help me understand one step in the constructing of product measure?

Given two measure space (X, A, $\mu$), (Y, B, v), let $\{A_k * B_k\}_{k=1}^{k=\infty}$ be a countable disjoint collection of measurable rectangles whose union also is a measurable rectangle A * B. ...
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### CLT for continuous functions of random variables

Let $(X_i)$ be a collection of zero mean, unit variance, real valued random variables (I do not assume that they are iid). Let $\mathcal H$ be a separable RKHS with a bounded kernel $k(x,y)$. Suppose ...
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### Why is Caratheodory's characterization of measurability important?

My professor repeatedly emphasizes the importance of Caratheodory's theorem about characterization of measurability, but I don't get why it's so important. As far as I remember, I have never used this ...
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### Algebra on infinite sets is not closed under countable union

this proof came up during self study, rather than looking at the other proofs out there, I would like to correct if necessary the following one... Let $X$ be an infinite set and $A$ the collection of ...
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### An uncountable chain of equivalence relations

First, an example: We know that, for two real valued, Lebesgue-integrable functions, the relation "equals almost everywhere" is an equivalence relation. In particular, if $f_0$ is Lebesgue-integrable, ...
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### Natural filtration of martingales

I don't quite understand what the natural filtration really is. Imagine e.g. a sequence of independent and identically distributed random $N(0,1)$ variables. What is their natural filtration, and how ...
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### Measure theory: upper bound for a particular set

I have the following problem: consider $A_1,...,A_N$ Borel set on [0,1] with measure greater than 1/2. For every a real number between $0$ and $\frac{1}{2}$, consider the set $E_a$={$x$| $x\in A_j$ ...
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### Is $\iint \dfrac{1}{z} dxdy\neq 0$?

I am trying to solve an exercise and at some point I came accross the integral $$\iint_L \dfrac{1}{z} dxdy,$$ ($z=x+iy$) where $L\subset \mathbb{C}$ is a compact set with positive two-dimensional ...
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### Is Bayesian Association mathematically rigorous?

Introduction. This question is based on the Ph.D. thesis of B.T. Vo, which can be found in this website ("Papers" section). More specifically, in the introduction of the Ph.D. thesis, at page 8, there ...
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### Function in indicator function and integral.

I have the following expression $$\int 1_{ t^{-1}(A)}(x) e^{t(x)} d\mu(x)$$ How do I express this integral in terms of $t(\mu)$? Specifically, what do I do about that indicator function which also ...
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### Determining null sets with Tonelli's theorem

How can I show that the diagonal $D=\{(x,y)\in\mathbb R^2\vert x=y\}$ is a Lebesgue-nullset in $\mathbb R^2$ by utilizing the theorem of Tonelli? My solution so far, but it doesn't seem quite right: ...
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### Computing a three-dimensional Lebesgue measure of a bounded set

How can I compute the three-dimensional Lebesgue-measure of the set $A$ which is bounded by the areas $x+y+z =6$, $x=0$, $z=0$ and $x+2y=4$? A hint on how I solve problems like this in general would ...
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### Can anyone help understand one step in the proof of a theoeam related to the norm of linear functional?

Proposition 2 Let E be a measurable set, 1 < p < oo, q the conjugate of p, and g belong to Lq(E). Define the functional T on $L^p(E)$ by $$T(f)=\int_Egf \ for\ all\ f\in L^p(E)$$ Then T is a ...
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If $\mu =\mu_1 + i \mu_2$. is a complex measure and $\mu_1$ and $\mu_2$ are positive and finite. What is the total variation $\left|\mu\right|$ in this two cases? 1) $\mu_1 \perp \mu_2$. 2) $\mu_1 =... 1answer 31 views ### Why do we have$\bigvee_{i=-\infty}^{\infty}T^i\mathcal{A}=\mathcal{B}$? Let$Y=\left\{0,1,\ldots,k-1\right\}, X=\prod_{-\infty}^{\infty} Y$and let$T$be the left shift. Let$A_i=\left\{\left\{x_k\right\}: x_0=i\right\}, 0\leq i\leq k-1$. Then$\xi=\left\{A_0,\ldots,A_{k-...
Exercise 16 - Let $(X,M,\mu)$ be a measure space. A set $E\subset X$ is called locally measurable if $E\cap A\in M$ for all $A\in M$ such that $\mu(A) < \infty$. Let $\tilde{M}$ be the collection ...
### A confusing step in the proof showing $\lambda(E)=\int \varphi\,\chi_E\, d\mu$ is a measure on $\bf X$
The proof I do not understand comes from the text The Elements of Integration by Bartle on page 28. It goes as follows: Observe that \varphi\,\chi_E=\sum_{j=1}^{...