# Tagged Questions

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### Extension of Premeasures

Here, a premeasure is a countably additive set function whereas a measure is one acting on a sigma-algebra. Not every positive premeasure admits an extension to a positive measure as the following ...
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### Does the class of all finite unions of closed-open intervals on $\mathbb{R}$ form a ring sets?

Does the class of all finite unions of closed-open intervals on $\mathbb{R}$ form a ring on sets? By a closed-open interval , I mean an interval of the form $[x,y)$ A ring of sets is a non-empty ...
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### Measures: Sequential Continuity

Disclaimer: This thread is meant as record and written in Q&A style. Let $\Omega$ be a finite measure space $\mu(\Omega)<\infty$. It is well known that a measure is continuous from above as ...
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### Vitali Set: Inner Measure vs. Outer Measure

Context Nonlinearity in general of the Lebesgue integral for nonmeasurable functions reduces in some sense to inner and outer measure of nonmeasurable sets: ...
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### How can I prove that $\{ \ (x,y)\in \mathbb R^2 : x >0, 0\le y \le 1/x \ \} \in \mathcal B(\mathbb R^2)$ is a Borel-set in $\mathbb R^2$?

How can I prove that $\{ \ (x,y)\in \mathbb R^2 : x >0, 0\le y \le 1/x \ \} \in \mathcal B(\mathbb R^2)$ is a Borel-set in $\mathbb R^2$ ? I have tried to construct this set from countably union ...
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### Let $X$ be a non-empty set, $A \subseteq X$. Decide the set $\mathcal M(\mathcal{E})$ of $\mathcal{E}$-$\mathcal B(\mathbb R)$-measureable functions.

Let $X$ be a non-empty set, $A \subseteq X$. Decide the set $\mathcal M(\mathcal{E})$ of $\mathcal{E}$-$\mathcal B(\mathbb R)$-measureable functions $f: X \rightarrow \mathbb R$ in each of the ...
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Suppose $\mu : \epsilon \rightarrow [0,\infty]$ is a measure from the $\sigma$-algebra $\epsilon$ into extended $\mathbb R$. How do I see $$\mu\left(\bigcap_{n \in \mathbb N} B_n\right) = ... 1answer 27 views ### Let \epsilon = \{A \subseteq \mathbb R : A \ \text {or} A^C \ \text {is countably}\}. I want to show \epsilon \subseteq \mathcal B(\mathbb R). Let \epsilon = \{A \subseteq \mathbb R : A \ \text {or} A^C \ \text {is countably}\}. I want to show \epsilon \subseteq \mathcal B(\mathbb R). As you probably already know \epsilon is a ... 1answer 70 views ### Is the union of an infinite sequence of finite sets an infinite set? Given the following question: "Let X be an infinite set and define \mu^* on \mathcal P \left({X}\right) by \mu^*(A) = 0 if A is finite and \mu^*(A)=1 if A is infinite. Is \mu^* an outer ... 1answer 11 views ### Let X \neq \emptyset and B \subseteq X. Show \epsilon_B := \{A \subseteq X : B \subseteq A \lor B \subseteq A^C \} is a \sigma-algebra. Let X \neq \emptyset and B \subseteq X. Show the system \epsilon_B := \{A \subseteq X : B \subseteq A \lor B \subseteq A^C \} is a \sigma-algebra in X. I've already proved the two ... 0answers 19 views ### showing that the sets (Banach-Tarski-ish) which comprise S^1 are disjoint Let S^1 be the unit circle and consider S^1 = \cup_{q \in \mathbb{Q}} A_q where the sets A_q are constructed as follows: Define the equivalence relation z \sim w if for z = e^{i\alpha}, w = ... 1answer 39 views ### Tails sets are Borel I am trying to proof a particular case of Kolmogorov's law in the set E of infinite binary sequences. Eventually, I'm supposed to prove that a certain type of subsets of this set is in the Borel sigma ... 2answers 31 views ### Measures: Sigma-Additivity vs. Continuity Let R be a ring of sets that contains the empty set and \mu be a positive and finite set function on R. If \mu is countable additive, then it is continuous from below and above:$$A_n\uparrow ...
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Disclaimer: Though this thread is written in a Q&A style any new thoughts are really welcome! What reasons are there to restrict measures to countable additivity rather than uncountable ...
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### How to prove the following defined collection is a sigma algebra?

Let $\mu$ and $\lambda$ be two measures on a $\sigma$-algbra $\mathfrak{F}$ on $\Omega$, such that $\mu (A)=\lambda(A)$ for any $A\in \mathfrak C$, where $\mathfrak C\subset\mathfrak{F}$ is a ...
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### How many algebras of subsets of $X$ contain exactly four elements?

Let X be a set with five elements. How many algebras of subsets of X contain exactly four subsets? Well $\emptyset, X$ must be in any algebra of subsets of $X$ so that means we have to find two more ...
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### Non-section representation of an intersection of sets

Let $X,\bar X,Y$ be arbitrary sets and $A\subseteq X\times Y$, $\bar A\subseteq \bar X\times Y$ be arbitrary as well. Denote: $$A_x :=\{y\in Y:(x,y)\in A\}$$ and similarly for $\bar A$. Consider a ...
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Let $\tau$ be a semialgebra of subsets of $\Omega$ and let P: $\tau\rightarrow [0,1]$, with $P(\Omega)=1$, and it satisfies finite additivity: $P\big(\bigcup_{i=1}^{n}D_i\big)=\sum_{i=1}^{n}P(D_i)$ ...
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### A basic property of the Lebesgue outer measure

If $G$ is a measurable set and satisfies $m^*(G)<\infty$, then for all $\varepsilon>0$ there exists a closed set $F\subset G$ such that $m^*(F)>m^*(G)-\varepsilon$ Edit: I know that: ...
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### Example of $\sigma$-algebra

I understood the definition of a $\sigma$-algebra, that its elements are closed under complementation and countable union, but as I am not very good at maths, I could not visualize or understand the ...
### Show that $\mathfrak{Z}$ is a semi ring
Consider measurable spaces $(\Omega_t,\mathcal{A}_t), t\in T$ ($T$ is any index set). With $\mathcal{E}(T)$ we the set of all finite, not-empty subsets of $T$. Show that  ...