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From: Dan Cass on 26 Jan 2010 01:11
> 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...
***If the collection is closed under union and
***under complement, then it's an algebra.
***DeMorgan then implies closed under intersection also.

> What happens if you take the finite sets, and their
> complements, and
> nothing else?
>
> --
> Arturo Magidin

The collection "FC" of finite or cofinite subsets
of the integers is an algebra.
[closed under union and under complement]

But it doesn't contain the set O of all odd integers.
Since for each odd integer z the set {z} is in FC,
we can say that if FC were a sigma algebra,
it should contain the (countable) union of the {z}'s
which is the set O of all odd integers.

This shows that FC is an algebra and not a sigma algebra.

That's probably what Arturo was getting at...
From: TCL on 26 Jan 2010 11:24
On Jan 25, 6:42 pm, Legendre <sinankap...(a)yahoo.com> wrote:
> I want to find a set which is algebra but not sigma-algebra.
>
> Thanks

Sets that are a finite disjoint union of sets of the form

(-infty,infty), (-infty, a), [a,b), [a,infty), emptyset

where a<b are real numbers.
-TCL
From: Arturo Magidin on 26 Jan 2010 11:32
On Jan 26, 10:11 am, Dan Cass <dc...(a)sjfc.edu> wrote:
> > 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...
>
> ***If the collection is closed under union and
> ***under complement, then it's an algebra.
> ***DeMorgan then implies closed under intersection also.
>
> > What happens if you take the finite sets, and their
> > complements, and
> > nothing else?
>
> > --
> > Arturo Magidin
>
> The collection "FC" of finite or cofinite subsets
> of the integers is an algebra.
> [closed under union and under complement]
>
> But it doesn't contain the set O of all odd integers.
> Since for each odd integer z the set {z} is in FC,
> we can say that if FC were a sigma algebra,
> it should contain the (countable) union of the {z}'s
> which is the set O of all odd integers.
>
> This shows that FC is an algebra and not a sigma algebra.
>
> That's probably what Arturo was getting at...

It's almost certainly what Tim was getting at too...

--
Arturo Magidin
From: David C. Ullrich on 26 Jan 2010 11:51
In article
<1491419083.50072.1264462981695.JavaMail.root(a)gallium.mathforum.org>,
Legendre <sinankapcak(a)yahoo.com> wrote:

> I want to find a set which is algebra but not sigma-algebra.

People have given two simple examples. In case you want
infinitely many more: If S is any countably infinite collection
of subsets of any set X then the algebra generated by S is
countable, and hence cannot be a sigma-algebra since any infinite
sigma-algebra has cardinality at least c.

> Thanks

--
David C. Ullrich
From: Tim Little on 26 Jan 2010 18:06
On 2010-01-26, Dan Cass <dcass(a)sjfc.edu> wrote:
> For example the collection of finite subsets of the integers is not
> an algebra... so this isn't the example Tim has in mind.

Correct. That's why it was a hint, and not a solution on a silver platter.


- Tim
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