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Questions tagged [semialgebras]

A semialgebra on a set is class of subsets of the set. It contains the original set and the empty set. Further the class is closed under finite intersections and any difference of two sets belonging to it can be written as a finite union of mutually disjoint elements of it. It is used especially in the theory of measures and probabilities.

A semialgebra on a set is class of subsets of the set. It contains the original set and the empty set. Further the class is closed under finite intersections and any difference of two sets belonging to it can be written as a finite union of mutually disjoint elements of it. It is used especially in the theory of measures and probabilities.

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Question about the definition of a semialgebra

This question has been asked here Question about definition of Semi algebra The OP unfortunately has selected an incorrect answer and no agreed upon correct answer has been given. The most upvoted answer finishes by saying he would be interested if…
Stan Shunpike
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Proving a semialgebra

The sets of the form of all $(a, b]$ intervals in $(0, 1]$ is given to be a semialgebra. If you take an inf intersection of $(\frac{a -1}{n}, b]$, for $n \to \infty$, for some valid $a$ and $b$ in $(0,1]$ you'll get the closed set $[a, b]$. This is…
Dilan
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Countable Subadditivity on a Semialgebra

I was reading "Measure Theory and Probability Theory" by Krishna B. Athreya and, in chapter 1, the subject of measures on semialgebras is brought up. In particular he introduces the following definition: Given a measure $\mu$ on a semialgebra…
Giordano Ribeiro
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On the definition of semi-algebra

I found two definitions in some textbooks. One is defined as follows: A collection $\mathscr{C}$ of subset of $X$ is called a semi-algebra if (a) $\emptyset \in \mathscr{C},$ (b) if $A, B\in \mathscr{C},$ then $A\cap B\in \mathscr{C},$ (c) for $A\in…
ljjpfx
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Question on semialgebra

This exercise comes from A First Look At Rigorous Probability (Exercise 2.7.19): Let $\Omega$ be a finite non-empty set, and let $\mathcal{J}$ consist of all singletons in $\Omega$, together with $\emptyset$ and $\Omega$. Show that $\mathcal{J}$ is…
user1691278
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Definition of a Semi Algebra

I am self-studying a book on Measure Theory and Probability Theory by Lahiri and Athreya. I encountered the following definition of a Semi-Algebra. Let $Ω$ be a nonempty set and let $P(Ω)$ be the power set of $Ω$. A class $C⊂P(Ω)$ is called a…
Jim Allyson Nevado
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Intersection of a semi-field and a subset is again, a semi-field

Let $S$ be a semi-field of subsets of $\Omega$ and $E$ be any subset of $\Omega$. Then, show that $S\cap E$ is a semi-field of subsets of $E$. I simply need to prove two properties for the collection of subsets of $E$, $S\cap E$: closed under…
zaira
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Semi-algebra generating $\mathscr{B}(\mathbb{R})$

I would like to know if the collection $\mathcal{S}$ consisting of intervals $[a,b)$ with $a\leq b$, $a,b\in\mathbb{R}$, is a semi-algebra. I'm familiar with the standard definition: $\emptyset\in\mathcal{S}$ $A,B\in\mathcal{S} \Rightarrow A\cap…
Oskar Vavtar
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The collection of half open half closed interval with empty set in $\mathbb{R}$ generated a semi-algebra

This question is about Durrett Edition 5 Example 1.1.8, in which he claims that Let $\Omega=\mathbb{R}$, and $\mathcal{S}=\mathcal{S}_{1}$ then $\overline{\mathcal{S}}_{1}=$the empty set put all sets of the form $\bigcup_{k=1}^{n}(a_{k}, b_{k}]$,…
JacobsonRadical
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How do we prove that all intervals contained in [0,1] is a semi-algebra?

Well, I know the definition of semi-algebra but I cannot prove that all intervals contained in [0,1] is a semi-algebra
antonis34
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Symbol in Theorem A.1.1 in Durrett's Probability (4th edition)

The highlighted symbol is found in Durrett's "Probability: Theory and Examples" (4th edition). What does it mean?
Incognito
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Semigroups and measure

A semialgebra $\mathcal S$ is a Set of subsets such that: $\emptyset, X \in\mathcal S$ where $X$ is the Universe. If $A,B \in\mathcal S$, then $A \cap B \in\mathcal S$ If $A \in\mathcal S$, then $A^c = A_1 \cup A_2 \cup ...... \cup A_n$ where $A_i…
S. Cow
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