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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:

  • $\emptyset\in\mathcal{S}$
  • $A,B\in\mathcal{S} \Rightarrow A\cap B\in\mathcal{S}$
  • $\forall A \in \mathcal{S}: A^\mathrm{c} = \bigcup_{i=1}^n E_i$ with $\{E_1,\ldots,E_n\}\subset \mathcal{S}$, disjoint.

The above mentioned collection does not satisfy the third condition. However, here https://math.byu.edu/~bakker/Math541/Lectures/M541Lec11.pdf I found an alternative definition of semi-algebra with which $\mathcal{S}$ complies (see the example under the definition).

Is this a more general definition of semi-algebra?

Ramiro
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Oskar Vavtar
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  • Assuming that $\mathcal{S}$ is not empty, the definition in the pdf file is actually the definition of semiring of sets, which is more general than the definition of semi-algebra. See, for instance, https://en.wikipedia.org/wiki/Ring_of_sets#Related_structures – Ramiro Oct 21 '22 at 11:24
  • The collection $\mathcal{S}$ consisting of intervals $[a,b)$ with $a\leq b$, $a,b\in\mathbb{R}$, is not a semi-algebra, but it is a semiring (of sets). – Ramiro Oct 21 '22 at 11:33
  • @Ramiro I understand. I was trying to use $\mathcal{S}$ in the standard ergodic theoretic application, where you "check" a certain property just for a generating semi-algebra. Would you happen to know if you could do a similar thing with a semiring in the $\sigma$-finite setting? – Oskar Vavtar Oct 21 '22 at 12:10
  • Yes. If $\mathcal{S}$ is a semiring of subsets of $\Omega$ and $\Omega$ can be covered by a countable collection of elements in $\mathcal{S}$, then $\Sigma$, the $\sigma$-ring generated by $\mathcal{S}$, is in fact a $\sigma$-algebra. In other words, you can replace $\mathcal{S}$ by $\mathcal{A}$, the smallest semi-algebra containing $\mathcal{S}$ and then the $\sigma$-algebra generated by $\mathcal{A}$ will be the same $\Sigma$ (the $\sigma$-ring generated by $\mathcal{S}$). – Ramiro Oct 22 '22 at 00:58

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