What is a Borel set?
Konrad Viltersten
the definition it is a bunch of sets included in sigma(G),
where G is the class of all open subsets in R.
What does it mean? Is it possible to give humanly
graspable examples of such G? How many different G's
can there exist (for the R-axis)?
My guess is that G is basically a discretization of the
continous R-axis where (almost) every point is assigned a
surrounding area around it, hence obtaining a kind of
"width". How wrong am i?
--
Vänligen
Konrad
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Sleep - thing used by ineffective people
as a substitute for coffee
Ambition - a poor excuse for not having
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gow...@hotmail.com
countable intersection of open sets. All F-sigma and G-delta sets are
Borel sets, but there are still more Borel sets. However, if you want
something "humanly graspable" maybe those will do.
Konrad Viltersten
According to what i understand here, a G-delta set will
only consist of one, single element, since it's an intersection
(as in numerus singularis), not a set of intersections.
It sounds a little bit suspicious so i'm guessing i missed
something here. Would it be possible to explicitly propose a
set of Borel type (i.e. using actual elements as if one were
talking to a retarded person)?
For instance, is the below a Borel set?
{ {0}, {1,2,3,4}, {1,2,4}, {3}, {4}, {1,2,3}, {3,4}, {1,2} }
I used
Omega = {1,2,3,4}
A = { {1,2,4}, {1,2,3,4} } = {elements consisting of: 1,2,4}
If so, can we see more examples on the form above (i'm
reminding of the retard-form)? If not, why not?
gow...@hotmail.com
Real Numbers is a set of numbers. Your example is a set of sets. Each
set is A is a Borel sete because they are finite sets and hence closed.
The set of all Borel sets contains all the F-sigma and all the G-delta
sets as well as infinitely many other sets.
José Carlos Santos
>> An F-sigma set is a countable union of closed sets. A
>> G-delta set is a countable intersection of open sets. All
>> F-sigma and G-delta sets are Borel sets, but there are still
>> more Borel sets. However, if you want something "humanly
>> graspable" maybe those will do.
>
> According to what i understand here, a G-delta set will
> only consist of one, single element, since it's an intersection
> (as in numerus singularis), not a set of intersections.
Take A_n = R for each natural _n_. Each A_n is an open set and therefore
their intersection (which happens to be R itself) is also a Borel set,
which clearly has more than one point.
> It sounds a little bit suspicious so i'm guessing i missed
> something here. Would it be possible to explicitly propose a
> set of Borel type (i.e. using actual elements as if one were
> talking to a retarded person)?
>
> For instance, is the below a Borel set?
> { {0}, {1,2,3,4}, {1,2,4}, {3}, {4}, {1,2,3}, {3,4}, {1,2} }
{0} is a Borel set, sint it's equal to the intersection of all sets of
the type ]-1/n,1/n[ (_n_ natural).
{1,2,4} is the intersection of the sets of the type
]1 - 1/n, 1 + 1/n[ U ]2 - 1/n,2 + 1/n[ U ]4 - 1/n,4 + 1/n[
(_n_ natural) and so on.
> I used
> Omega = {1,2,3,4}
> A = { {1,2,4}, {1,2,3,4} } = {elements consisting of: 1,2,4}
What do you mean by this? In your original post, you said that you were
working with "the class of all open subsets in R".
Best regards,
Jose Carlos Santos
Jules
Konrad Viltersten wrote:
> I'm trying to understand what a Borel set is. According to
> the definition it is a bunch of sets included in sigma(G),
> where G is the class of all open subsets in R.
>
> What does it mean? Is it possible to give humanly
> graspable examples of such G? How many different G's
> can there exist (for the R-axis)?
I do not quite understand your question about G. You said G was the
class of open sets. There are no examples of such G; there is only one
class that G can be. Are you looking for examples of Borel sets? If
so, here is something to keep in mind: Almost every set that you will
run into is a Borel set. It takes a certain amount of work to show
that there are some sets which are not Borel sets. Open sets, closed
sets, and sets which are countable unions or intersections of open sets
and closed sets are all Borel sets.
Chen
Is a "half open, half closed" set a Borel set or not?
José Carlos Santos
> I encountered the same problem.
Perhaps that you did not notice it, but you are replying to a post which
was posted here more than two years ago.
> Is a "half open, half closed" set a Borel set or not?
Do you mean something like { x | 0 <= x < 1 }? Of course it is. It is
the intersection between an open set and a closed one.
David C. Ullrich
wrote:
>I encountered the same problem.
>
>Is a "half open, half closed" set a Borel set or not?
If you're asking about a half-open _interval_, like
[a,b), the answer is yes (hint: it's a countable union
of closed sets).
If you really meant to say "set" instead of interval
the answer is "exactly what do you mean by
'half-open, half-closed set'?".
David C. Ullrich
Warren
Let us consider a topological space [0,1]. What will be the Borel sets/ sigma algebra of this space ?
Gc
> Can anyone explain Borel sets or Borel algebra with the following example.
>
> Let us consider a topological space [0,1]. What will be the Borel sets/ sigma algebra of this space ?
The sets that belong to the intersection of all sigma algebras that
contain all the open sets.
G. A. Edgar
<d478ef76-1081-4913...@l13g2000yqb.googlegroups.com>, Gc
<gcu...@hotmail.com> wrote:
Maybe a less elliptic answer: the smallest sigma-algebra that contains
all open sets.
Warren's answer is to the next question: How do you know there is a
smallest one?
So, for example, any open set is a Borel set. Any closed set is a
Borel set. Any G-delta is a Borel set. Any F-sigma is a Borel set.
There are more complex ones as well, of course, but start with these.
--
G. A. Edgar http://www.math.ohio-state.edu/~edgar/
Zdislav V. Kovarik
The lengthy answer to the question "how is it built from below?" requires
countable ordinal numbers.
For n=0, collect all open sets.
For a successor countable ordinal n=m+1, collect all countable unions of
all countable intersections of sets collected at step m.
For a limit countable ordinal n, collect all sets obtained at steps
preceding n.
If I remember correctly, on the real line the (transfinite) sequence so
indexed is strictly increasing. I do not have a quotation, though. Anyone?
Cheers, ZVK(Slavek).
Dave L. Renfro
> If I remember correctly, on the real line the (transfinite)
> sequence so indexed is strictly increasing. I do not have
> a quotation, though. Anyone?
Lebesgue proved this in 1905 using a Cantor diagonal argument
applied to universal sets for the alpha'th Borel level
('alpha' being any given countable ordinal). See Theorem 2.5
on pp. 11-12 of
Arnold W. Miller's 6 January 1994 manuscript "Descriptive
Set Theory and Forcing"
http://arxiv.org/PS_cache/math/pdf/9401/9401202v1.pdf
Dave L. Renfro