# Tagged Questions

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

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### Two norms of stochastic kernels

Let $(E,\mathscr E)$ be a measure space and $P:E \times\mathscr E\to [0,1]$ be a stochastic kernel - i.e. $$P(x,A)\in [0,1]$$ for any $x\in E$ and $A\in \mathscr E$. On a set $b\mathscr E$ of ...
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### Extending a set of complex Borel measures defined on subsets to the whole space

Continuing my work through Folland, trying to prove the following (Chapter 7 #22): Added: *Let $X$ be a locally compact Hausdorff space.* Let $\{f_\alpha\}_{\alpha\in A}$ be a subset of $C_0(X)$ ...
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### An approach to Borel-Cantelli for the $l^p$

Let $\mu$ be a non-negative measure and $\{E_{k}\}$ a sequence such that $\sum \mu(E_k)^p<\infty$ then show that $F=\lim \ \sup E_{k}=\cap_{k=1}^{\infty}\cup_{n\geq k}E_n$ has $\mu$ measure zero.
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### A question about the Lebesgue-Stieltjes measure of the Cantor function

This question is a follow-up to the post "Calculating a Lebesgue integral involving the Cantor Function." Let $\varphi: [0,1] \rightarrow [0,1]$ be the Cantor (ternary) function, and let $m_\varphi$ ...
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### Weak convergence of measures and convergence of “almost densities”

Let $f_i$ be a sequence of smooth functions on $S^2$ such that the measures $\mu_i=f_i \;d\mathrm{vol}_{S^2}$ converge weakly to $d\mathrm{vol}_{S^2}$. Now suppose $\epsilon_i$ is a sequence going to ...
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### Understanding no free lunch theorem

From Wikipedia: $Y^X$ is the set of all objective functions $f$:$X$→$Y$, where $X$ is a finite solution space and $Y$ is a finite poset. The set of all permutations of $X$ is $J$. A random ...
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### Kolmogorov’s example of a measurable function not (generally) differentiable

In [1, page 7], the author says. Kolmogorov showed that if the function $$f(x) = \sum_{n=1}^{\infty} \frac{\cos 3^n x}{3^n}$$ has a finite or infinite generalized derivative on a set of positive ...
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### A consequence of the Fubini-Tonelli theorem?

Tonelli-Fubini Theorem. Let $(\mathbb{X},\mathscr{X},\mu)$ and $(\mathbb{Y},\mathscr{Y},\nu)$ be probability spaces and let $\mathscr{Z}$ be the $\sigma$-field product i.e. the $\sigma$-field ...
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### Generated Sigma-algebra example

I know that a sigma algebra generated by a subset a, $\sigma(a) = \{\emptyset,a,a^c,E\}$. But what about $\sigma({a,b})$? Would it be $\{\emptyset,a,a^c, b, b^c,E\}$?
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### reference for “compactness” coming from topology of convergence in measure

I have found this sentence in a paper of F. Delbaen and W. Schachermayer with the title: A compactness principle for bounded sequences of martingales with applications. (can be found here) On page 2, ...
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### Existence of a structure-preserving mapping between two spaces?

I have some questions, but not sure if they are meaningful: Suppose $X$ and $Y$ are two arbitrary measurable spaces. Does there exist a measurable mapping from $X$ to $Y$? Suppose $X$ and $Y$ are ...
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### from almost everywhere convergence to uniform convergence

Suppose $\mu$ is sigma-finite measure on a space $X$, and $f_{n}$ converge to $f$ almost everywhere. Show that there exists measurable sets $E_{n}$ where $\mu ( \cap E_{n}^{c}) = 0$ and $f_{n}$ ...
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### How to show that $\frac{f}{g}$ is measurable

Here is my attempt to show that $\frac{f}{g}~,g\neq 0$ is a measurable function, if $f$ and $g$ are measurable function. I'd be happy if someone could look if it's okay. Since $fg$ is measurable, ...
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### Separability of the set of positive measures

Let $X$ be a locally compact separable & metrizable space, and $M^{+}(X)$ its space of positive measures (i.e. positive linear forms on the space of continuous functions on $X$, continuous on each ...
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### Integral of sequence converges? [closed]

Suppose $(X, \mathcal A, \mu)$ is a measure space, $f$ and each $f_n$ is integrable and nonnegative, $f_n \to f$ almost everywhere, and $\int_X f_n \to \int _X f$. Does it necessarily follow that for ...
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### Sequence of nonnegative $f_n$ tending to $0$ pointwise where $\int f_n \to 0$, but there's no integrable function where $f_n \le g$ for all $n$?

What is an example of a sequence of nonnegative functions $f_n$ tending to $0$ pointwise such that $\int f_n \to 0$, but there is no integrable function such that $f_n \le g$ for all $n$?
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### What is an example of lower semicontinuous functions not satisfying this?

Let $X$ be a locally compact Hausdorff space and $\mu$ be a Radon measure on $X$. Let $u:X\rightarrow [0,\infty]$ be a lower semicontinuous function such that $\int_X u d\mu <\infty$. Then, does ...
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### Can we approximate integrable functions by finite-valued upper,lower semicontinuous functions?

Let $X$ be a locally compact Hausdorff space and $\mu$ be a Radon measure on $X$ and $f\in L^1(\mu)$ be real-valued and $\epsilon>0$. Then, by Vitali-Caratheodory theorem, there exist upper ...
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### Exchanging supremum and conditional expectation

I've come across a problem which seems similar to this but quite different and can't find a way of going around it. I am working with a continuous process $Y_t$ generating the filtration $(F_t)$ and ...
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Let $u$ be a subharmonic function on the disk $\Delta$ centered at the origin with radius $\rho$. $u$ is not identically $-\infty$. Define $f:\Delta\times [0,2\pi]\rightarrow [-\infty, \... 0answers 30 views ### Why$\lim_{x\to \infty }Mf(x)=0$where$Mf$is the Hardy-Littlewood maximal function. In a course it is written that since$Mf(x)$decay to zero at infinity, the measure or$\left\{x\in\mathbb R^n\mid\left|Mf(x)\right|>\lambda\right\}$is finite. I was looking for such a result on ... 0answers 26 views ### If$\mu(E)\geqslant 0$is it true that$E\in \mathfrak{M}$? Suppose$(X,\mathfrak{M},\mu)$be a mesure space. Let$E$such that$\mu(E)\geqslant 0$. Can we conclude that$E\in \mathfrak{M}$? I think YES because$\mu$is the set function with domain$\mathfrak{...
Suppose that $A$ and $B$ are compact metric spaces. Let $f:A\times B\to B$ be a Borel measurable map (in the sense that for every Borel set $S\subseteq B$, $f^{-1}(S)$ belongs to the $\sigma$-algebra ...
### Haar measure, can image of modular function be any subgroup of $(0,\infty)$?
It is easy to find examples of locally compact second countable Hausdorff topological groups $G$ whose modular function $\Delta$ has image $\{1\}$ or $(0,\infty)$. Are there groups $G$ of this kind ...