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# Sigma-algebra help watch

1. I'm not sure how to begin with these. I know the definition of a -algebra of events and the 3 conditions, but I can't seem to see how I can use it here, unless there's some other simple thing my brain keeps missing out on.
2. The question is, I think, missing a few definitions from the top, however, assuming F and G are both sigma-algebras, this is basically saying:

(ii) Given that F and G are sigma-algebras, show that F intersect G is a sigma-algebra.
(iii) Given that F and G are sigma-algebras, show that F union G is not necessary a sigma-algebra.

Essentially, for (ii), show that H (the intersection of F and G) contains the empty set, is closed under complementation, and closed under countable unions and countable intersections, given that F and G satisfy these properties.

For (iii), you must build a counterexample: find and state specific F and G which are sigma-algebras (prove this) but that F union G is not a sigma-algebra (contradict one of the three parts of the definition).

Does this give you something to start from?

P.S. Sorry for lack of latex.
3. (Original post by RDKGames)

I'm not sure how to begin with these. I know the definition of a -algebra of events and the 3 conditions, but I can't seem to see how I can use it here, unless there's some other simple thing my brain keeps missing out on.
Assuming that F and G are sigma-algebras, you need to show, for example, that:

But

Now apply what it means for F,G to be sigma-algebras w.r.t complements.
4. (Original post by President Snow)
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(Original post by atsruser)
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Ah thanks, finally clicked I think. It is indeed given that are -algebras - forgot to mention, and use this fact properly. Here's what I did:

Part (ii)
Spoiler:
Show

We know that and thus hence

If then and , but then we also have and , thus

If then and this implies that . Similarly, we have . Thus

This hence proves that forms a -algebra of events.

Part (iii)
Spoiler:
Show

Say and define and then we have but thus it is not a algebra

Does that seem right? I have doubts on part (iii) - can I be more general here?
5. (Original post by RDKGames)
Ah thanks, finally clicked I think. It is indeed given that are -algebras - forgot to mention, and use this fact properly. Here's what I did:

Part (ii)
Spoiler:
Show

We know that and thus hence

If then and , but then we also have and , thus

If then and this implies that . Similarly, we have . Thus

This hence proves that forms a -algebra of events.

Part (iii)
Spoiler:
Show

Say and define and then we have but thus it is not a algebra

Does that seem right? I have doubts on part (iii) - can I be more general here?
This is good - students often struggle coming up with a counterexample for part (iii) but you got there [this question is commonly used in all Year 1 Probability courses].

I have a few points to note:

If then .

There's no need to go around the houses using F and G here [although you do need to do closure under countable unions similarly].

Also, do not write . This is incorrect. a and b are elements of A and B, not necessarily sets. You cannot union them. What you mean is .

Next, spell out for the examiner that you know F and G are sigma-algebras, even if you don't prove it. Something like this:

Say and define and , which we note are both sigma-algebras, then we have but thus is not a algebra.

You don't need to go any more general than this. It's already fairly general and generality is not required in a counterexample.

Well done.

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