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From: Legendre on 25 Jan 2010 08:42
I want to find a set which is algebra but not sigma-algebra.

Thanks
From: Tim Little on 26 Jan 2010 00:05
On 2010-01-25, Legendre <sinankapcak(a)yahoo.com> wrote:
> I want to find a set which is algebra but not sigma-algebra.

So, you want an algebra where all finite unions are in the set, but at
least some countably infinite unions are not. Hint: every finite
union of finite sets is finite.


- Tim
From: Dan Cass on 25 Jan 2010 21:57
> On 2010-01-25, Legendre <sinankapcak(a)yahoo.com>
> wrote:
> > I want to find a set which is algebra but not
> sigma-algebra.
>
> So, you want an algebra where all finite unions are
> in the set, but at
> least some countably infinite unions are not. Hint:
> every finite
> union of finite sets is finite.
>
>
> - Tim

I think an algebra is closed under both finite
unions and complements.

For example the collection of finite subsets of
the integers is not an algebra... so this isn't
the example Tim has in mind.

Off hand I can't come up with an algebra that isn't
a sigma algebra...
From: Jussi Piitulainen on 26 Jan 2010 08:45
Dan Cass writes:
> > On 2010-01-25, Legendre <sinankapcak(a)yahoo.com>
> > wrote:
> > > I want to find a set which is algebra but not sigma-algebra.
> >
> > So, you want an algebra where all finite unions are in the set,
> > but at least some countably infinite unions are not. Hint: every
> > finite union of finite sets is finite.
>
> I think an algebra is closed under both finite unions and
> complements.
>
> For example the collection of finite subsets of the integers is not
> an algebra... so this isn't the example Tim has in mind.
>
> Off hand I can't come up with an algebra that isn't a sigma
> algebra...

I think I've seen sets called co-finite, possibly without the hyphen,
when their complements are finite. Perhaps the finite and co-finite
subsets of some infinite set might work.
From: Arturo Magidin on 26 Jan 2010 10:50
On Jan 26, 6:57 am, Dan Cass <dc...(a)sjfc.edu> wrote:
> > On 2010-01-25, Legendre <sinankap...(a)yahoo.com>
> > wrote:
> > > I want to find a set which is algebra but not
> > sigma-algebra.
>
> > So, you want an algebra where all finite unions are
> > in the set, but at
> > least some countably infinite unions are not.  Hint:
> > every finite
> > union of finite sets is finite.
>
> > - Tim
>
> I think an algebra is closed under both finite
> unions and complements.
>
> For example the collection of finite subsets of
> the integers is not an algebra... so this isn't
> the example Tim has in mind.

That's not the only thing they have of course; but if you consider the
smallest algebra that contains finite sets...

What happens if you take the finite sets, and their complements, and
nothing else?

--
Arturo Magidin
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