# Tagged Questions

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

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### Example of a set and two $\sigma$ algebras such that union is not a $\sigma$-algebra

What is an example of a set $X$ and two $\sigma$-algebras $\mathcal{A}_1$ and $\mathcal{A}_2$, each consisting of subsets of $X$, such that $\mathcal{A}_1 \cup \mathcal{A}_2$ is not a $\sigma$-algebra?...
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### Example of a set and monotone class where monotone class is not a $\sigma$-algebra?

What is an example of a set $X$ and a monotone class $\mathcal{M}$ consisting of subsets of $X$ such that $\emptyset \in \mathcal{M}$, $X \in \mathcal{M}$, but $\mathcal{M}$ is not a $\sigma$-algebra?
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### Weak measurability of a set-valued map

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 ...
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### Is $\pi^1:C_c(G)\rightarrow \operatorname{End}(H)$ a homomorphism of the convolution algebra when $G$ is not unimodular?

Let $G$ be a Hausdorff locally compact group and $H$ a Banach space. Let $\pi:G\rightarrow \operatorname{GL}(H)$ be a representation and define $$\pi^1(\phi)v = \int_G\phi(x)\pi(x)vdx$$ for $v\in H$ ...
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### Construction of Lebesgue measure in Rudin's RCA book

This theorem from Rudin's RCA book. Here's one moment from it's proof which seems to me very weird. Rudin states that equality $\lambda(E)=m(E)$ holds for all Borel sets. But I think that it's ...
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### Lebesgue outer measure is countably subadditive but not finitely additive proof

I have read all the Qs on this but couldn't find a clear proof. How can I prove that Lebesgue's outer measure is not finitely additive? Thanks! Edit: I understand I must show that the measure of the ...
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### convergence in distribution in Banach spaces

We let $\Omega$ be a compact metric space and consider $C(\Omega)$ to be the space of all continuous functions on $\Omega$. The dual space of $C(\Omega)$ can be seen as the set of all signed borel ...
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### Real Analysis, Folland Proposition 2.29 Modes of Convergence

Background Information: $f_n\rightarrow f$ in $L^1$ $\Leftrightarrow$ $\forall\epsilon > 0,\exists N$ $\forall n\geq N$ $\int |f_n - f| < \epsilon$ A sequence $\{f_n\}$ of measurable complex-...
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### If $X_n \stackrel{p, quickly}{\to} X$, then $X_n \to X$.

Probability with Martingales: Without using hint, can I just do something like this: http://math.stackexchange.com/a/1538503/140308 ? With using hint: By continuity of probability, I think ...
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### 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 ...