algebra but not sigma-algebra
Legendre
Thanks
Tim Little
> 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
Dan Cass
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...
Jussi Piitulainen
> > On 2010-01-25, Legendre <sinan...@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.
>
> 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.
Arturo Magidin
> > On 2010-01-25, Legendre <sinankap...@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
Dan Cass
> > > On 2010-01-25, Legendre <sinankap...@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...
***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...
TCL
> 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
Arturo Magidin
It's almost certainly what Tim was getting at too...
--
Arturo Magidin
David C. Ullrich
<1491419083.50072.12644...@gallium.mathforum.org>,
Legendre <sinan...@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
Tim Little
> 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
Zdislav V. Kovarik
On Mon, 25 Jan 2010, Legendre wrote:
> I want to find a set which is algebra but not sigma-algebra.
>
> Thanks
On the real line, try the algebra A generated by all intervals
of the type [a,b). Can you find the interval (0,1) in it?
(It is in the sigma-algebra generated by A.)
(And you can do the proving yourself.)
(Or: in the positive integers, the algebra generated by finite subsets...)
Cheers,
ZVK(Slavek).
