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I have the following question:

$\textbf{Question}:$ Let $\mathcal{F}_1$ and $\mathcal{F}_2$ be two algebras. Is $\mathcal{F}_1 \cap \mathcal{F}_2$ a $\sigma$-algebra?

I believe the answer is no. I know the definition of an algebra and the definition of $\sigma$-algebra.

I'm assuming I need to show that the intersection of the finite unions of each algebra is finite, but then I get stuck trying to work out how to show it is not a $\sigma$-algebra.

All help is appreciated.

Many thanks,
John

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  • $\begingroup$ In general: if you are having troubles disproving something, try to prove the opposite. Maybe your intuition is misleading you. $\endgroup$ – quapka May 25 '15 at 8:12
  • $\begingroup$ @quapka Your comment is misleading actually... $\endgroup$ – GPerez May 25 '15 at 8:23
  • $\begingroup$ I do apologize. I've misread the question. I thought both $\mathcal{F}_1, \mathcal{F}_2$ are $\sigma$-algebras. Therefore I concluded, that it is indeed true. $\endgroup$ – quapka May 25 '15 at 21:15
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Some steps:

  • The intersections of two algebras $\mathcal F_1$ and $\mathcal F_2$ still is an algebra. Indeed, the whole set belongs to the intersection of the algebras, as well as the complement of an element of $\mathcal F_1\cap \mathcal F_2$. Stability by finite intersections also holds.

  • Therefore, the question is actually equivalent to the following one: is an algebra necessarily a $\sigma$-algebra? Indeed, if the answer to this question is yes, then you use the first step to get an affirmative answer to the initial question; if the answer is no, there is an algebra $\mathcal F_1$ which is not a $\sigma$-algebra, then pick $\mathcal F_2$ as the power set to get a counter-example.

  • If you consider the set of integers and the collection of subsets $A$ such that $A$ is finite or $\mathbf Z\setminus A$ is finite, you get a counter-example.

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  • $\begingroup$ Thanks Davide. I think I understand. I can show that the intersection of two algebras is an intersection, then using THAT algebra I can show that it fails to fulfil the third postulate of a sigma-algebra (i.e. union to infinity). Thanks. $\endgroup$ – John Smith May 25 '15 at 8:20

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