Re: Non-zero gaps between real numbers

On Dec 10, 3:01 am, David C. Ullrich <ullr...@xxxxxxxxxxxxxxxx> wrote:
On Sun, 9 Dec 2007 15:34:55 +0000 (UTC), Dave Seaman



<dsea...@xxxxxxxxxxxx> wrote:
On Sun, 09 Dec 2007 07:24:50 -0600, David C Ullrich wrote:
On Fri, 7 Dec 2007 13:01:15 +0000 (UTC), Dave Seaman
<dsea...@xxxxxxxxxxxx> wrote:

On Fri, 07 Dec 2007 06:18:13 -0600, David C Ullrich wrote:
On Thu, 6 Dec 2007 14:21:51 +0000 (UTC), Dave Seaman
<dsea...@xxxxxxxxxxxx> wrote:

On Thu, 06 Dec 2007 07:41:51 -0600, David C Ullrich wrote:
On Wed, 5 Dec 2007 15:32:39 +0000 (UTC), Dave Seaman
<dsea...@xxxxxxxxxxxx> wrote:

On Wed, 05 Dec 2007 06:43:11 -0600, David C Ullrich wrote:
On Tue, 4 Dec 2007 13:26:51 +0000 (UTC), Dave Seaman
<dsea...@xxxxxxxxxxxx> wrote:

On Tue, 04 Dec 2007 05:16:57 -0600, David C Ullrich wrote:
On Tue, 4 Dec 2007 04:09:32 +0000 (UTC), Dave Seaman
<dsea...@xxxxxxxxxxxx> wrote:

On Mon, 03 Dec 2007 17:25:30 -0600, David C Ullrich wrote:

The definition of the measure of the interval [a,b] is b-a.

No, that's not the definition. The definition involves first defining
outer measure and then demonstrating (using the Heine-Borel theorem) that
the outer measure of an interval is its length. Finally, one must shaw
that intervals are indeed measurable and therefore that the outer measure
is the measure.

Uh, thanks. We all knew that - this sort of technicality has
no real bearing here. In the context of the present discussion
there's no reason to assume that the word "measure" means the
same thing as it does in measure theory, seems better simply
to regard "measure" as a synonym for "length".

I know that *you* know that, but the OP specifically asked whether the
measure of an interval *follows from* the definition for measure, and you
answered that no, that *is* the definition for measure of an interval.
Wrong.

For one thing, if we tell VR that the measure of an interval is *defined
to be* its length, he is bound to consider the definition somewhat
arbitrary.

It _is_ somewhat arbitrary! In the measure-theoretic construction
you insist on alluding to there is somewhere an arbitrary definition
that the length of [a,b] is b-a. Or that the outer measure of a set
S is the inf of the sum of b_j - a_j, where (a_j,b_j) is a countable
collection of open intervals covering S.

Correct. But there is still the essential point that measure and length
are not the same thing, and there actually is something to prove when we
assert that m([0,1]) = 1.

_If_ the word "measure" means what it means in measure theory then
yes. But given the way VR sprays word salad around, concluding
that he's referring to what you and I call a "measure" just because
he uses that word seems simply ridiculous.

People have been urging VR to learn some measure theory. Some of his
questions seem to be prompted by measure theory. He certainly has some
misconceptions about what a measure is, but I choose to try to address
those misconceptions and not assume that he is deliberately misusing the
word.

Or whatever. There are many ways to "construct" Lebesgue measure -
in each such construction there _is_ an "arbitrary" b - a that
appears at some point (and if that is replaced by something
else then we get a measure other than Lebesgue measure).

Yes.

Hiding the "arbitrary" part in the middle of a complicated proof
and then suggesting that because the proof exists there's
nothing arbitrary about the b-a is disingenuous. There _is_
an arbitrary b-a involved.

Yes, because it's possible to define other measures on R and therefore
the choice of Lebesgue measure is somewhat arbitrary.

And hence claiming that there's nothing arbitrary about the fact
that the length of [a,b] is b-a, that it in fact follows from
deeper mathematics, is simply wrong.

I have made no such claim.

You said this, a few posts up:

"For one thing, if we tell VR that the measure of an interval is
*defined
to be* its length, he is bound to consider the definition somewhat
arbitrary. It's good to know that there is some deeper reason."

Look again at what you claim I said. In fact, I have not used the words
"nothing" and "arbitrary" together in a sentence at any time in this
thread,

I didn't say you had. Note tha lack of quotation marks in my
comment about what you said.

nor have I stated the equivalent in any other words.

Puhleaze. When you say "if we say [whatever], he is bound to consider
the definition somewhat arbitrary" as part of an explanation of
why we should not say [whatever] you're _not_ saying that
the definition is _not_ somewhat arbitrary?

Correct. I'm not.

Wow. Lemme see if I got this straight. We shouldn't say
[whatever] because that might give VR the idea that
the definition is somewhat arbitrary. The definition
_is_ somewhat arbitrary, but nonetheless, saying
we might give him the idea that it's somewhat
arbitrary is a valid reason why we must not say
[whatever].

The arbitrary part lies in the choice of Lebesgue
measure, as opposed to some other measure. However, the essential point
that I am trying to make about countable additivity is not limited to
Lebesgue measure. Notice that VR keeps asking why a union of
measure-zero sets can have positive measure. The answer to that question
does not depend on length.

If you're insisting that when you made that comment you were
_not_ claiming that the definition is _not_ somewhat arbitrary
then your comment makes no sense: If the definition is in fact
somewhat arbitrary then why would giving him the idea that it's
somewhat arbitrary be a bad thing?

Saying that m([0,1]) = 1 because it follows from the Heine-Borel theorem
is less arbitrary than saying that it's simply defined that way.

First, this is irrelevant to our current quibbles about who
said what. Second, while it may be less arbitrary, it's
also less honest.



Or maybe your point is some subtle distinction between
"the definition is not somewhat arbitrary" and
"there's nothing arbitrary about the definition".
If so you should explain - they seem to me to say
exactly the same thing.

Measure theory is not a deeper reason that the measure of an
interval is the same as its length - the reason that Lebesgue
measure was set up the way it is is that we _wanted_ the
measure of an interval to equal its length.

You said measure is *defined* to be the length. You then tried to
justify this by claiming that measure and length are the same thing. Was
your definition circular?

I didn't say that the measure was defined to be the length.
I said "The definition of the measure of the interval [a,b] is b-a."

Which is wrong.

Or right, depending on what we mean by the word "measure".

And also that it seemed clear to me that we were taking "measure"
and "length" to be synonymous. Making what I said equivalent to
"The definition of the length of the interval [a,b] is b-a."
Which is true. And not circular.

Wrong on both points.

In the first place, measure is not length.

I didn't say it was.

Yes, you did. You said (quoting here):

On Mon, 03 Dec 2007 17:25:30 -0600, David C Ullrich wrote:
The definition of the measure of the interval [a,b] is b-a.

That doesn't say measure is length.



Otherwise the measure of the
rationals would not be 0.

How do you explain to VR that measure is countably additive, but not
uncountably additive?

How do _you_ explain to _me_ why Lebesgue measure is
not uncountably additive? I'm not aware of any deep
reason for this fact, other than the "explanation"
"well, it's clear that it simply __isn't_."

I did explain it. The proof uses the definition of measure, plus the
Heine-Borel theorem and a bit of induction on the number of intervals in
the covering set. I mentioned this right up top, and you were
dismissive.

Seriously. It _is_ obvious that Lebesgue measure
is not uncountably additive, but what's the deeper
reason for that?

The last time I mentioned Heine-Borel, your response was "Uh thanks. We
all knew that - this sort of technicality has no real bearing here."

If it has no bearing here, then why are you asking about it now?

_I_ was talking about a different question at that time.

Anyway. So I haven't been paying attention. Exactly how is the
Heine-Borel theorem a "deeper reason" for the fact that
Lebesgue measure is not uncountably additive?

I
thought it was already established that you understood all that boring
stuff.

(No, the fact that there's no mention of uncounable additivity
in the axioms for measure theory is not anything like an
explantion of why Lebesgue measure is _not_ uncountably
additive. It's an explanation of why uncountable additivity
does not follow from the axioms of measure theory, not
an explanation for why it _is not_ uncountably additive.)

You have a short memory. I propose you go back and study the thread and
then get back to me.

************************

David C. Ullrich

The intervals' extents, the differences of their endpoints, defines
their measure, the measure function is defined in 1-D by the norm
which coincides with real value. Then, in terms of intervals not
including some non-empty countable subset of the interval, the subset
is deemed to have measure zero though it is non-empty. Then, the
difference in measure between the continuum segment, with its
intuitive measure being its length, and countably less subset is
considered zero, because that's to accommodate transfinite cardinals.
Indeed that is involved with the notion of infinitesimal quantities,
where generally zero is the only standardly real quantity described as
infinitesimal, even though infinitesimal means small and non-zero.

Standard measure is defined by line segments' extents. Most questions
of analytical interest are continuum problems, not about those defined
on only the rationals or irrationals, for example. There is the
notion of nonmeasurable sets, and they are described to exist because
otherwise there would exist some infinitesimal constant, eg Vitali's
c, the naturally infinite sum over which would be between 1 and 3.

The sum over the differentials d, Leibniz' d, in the unit interval, in
linear Leibniz' notation Int_0^1 1 dx, equals one, where the
antiderivative is x + C, and the definite integral evaluates to 1 - 0
= 1. The integral bar is an elongated S for summation, summation of
those differentials. For general nice continuous integrable
functions, for no finite difference is the integral correct, and for a
zero difference the integral is always trivially zero, so the true
differential is infinitesimal.

That's as well represented in the notions of probabilistic events that
are not impossible yet have probability zero, or uncertain with
probability one.

Ross

--
Finlayson Consulting
.

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