Re: obvious fact about volume
- From: David C. Ullrich <ullrich@xxxxxxxxxxxxxxxx>
- Date: Tue, 04 Oct 2005 08:18:02 -0500
On Mon, 03 Oct 2005 22:16:20 GMT, "NotP" <spam@xxxxxxxx> wrote:
>>
>>>I was worried that when you removed A_i, there might not be enough left in
>>>the sum
>>>to cover the reamaining A_j's.
>>
>> Not sure what you mean... Oh: You mean until you realized
>> what you just said you were worried that things might
>> overlap, so you'd need one A_j,k for part of two different
>> A_i's?
>>
>> After posting _I_ was worried that you were going to misread
>> my "And also for each i there is a _subset_ of the
>> A_j,k", since it wasn't all that well put. Of course I meant
>> a subset of the collection of all the A_j,k; if we had to
>> divide a single A_j,k into subsets the argument wouldn't
>> work.
>
>I did not misread what you said, I understood the idea.
>
>While you are here...do you have any helpful comments to make about Rudin's
>construction of Lebesgue measure?
I was puzzled by this at first, because I didn't recall Rudin
using anything like what's above in constructing Lebesgue
measure, I thought he used the Riesz Representation Theorem,
starting with the Riemann integral. Looking at the book,
I see that what you're really asking about is his
construction of the Riemann integral...
>I'm having trouble dealing with several aspects of it. For instance,
>vol(E_r) -> Vol(W) (p. 52). It seems intuitively true
>that if you fill up W with smaller and smaller boxes, the volume of the
>union will approach the volume of W.
The fact that vol(E_r) -> Vol(W) doesn't depend on what you
said, really, it's immediate from the definition of Vol and
the definition of E_r:
Take k = 2 again. Note that I haven't checked whether we're
talking about open or closed rectangles, etc, so you may
have to revise some of what's below. Say W = [a,b]x[c,d].
Then the construction of this and that shows that
E_r = [a',b']x[c',d'], where
a <= a' <= a + 2^{-r}
with similar inequalities for the b's, c's, and d's.
Those inequalities show that (b'-a')(d'-c') ->
(b-a)(d-c) as r -> infinity.
>But I find volume to be an unwieldy
>beast to work with, as demonstrated previously.
>or how about f < W implies that I(f) <= Vol(W), where I is the linear
>functional constructed in the proof.
Skimming through the proof I don't see exactly this assertion
anywhere. Of course the proof of the RRT shows that I(f)
<= m(W), and in the section you're asking about he proves
that m(W) = Vol(W).
But I really can't tell exactly what step you're wondering
about, and I don't feel like typing up a detailed explanation
of _every_ step you might find confusing. Tell me exactly
what step in the proof bothers you. (Um, also specify
which edition of the book you're reading, at least if
you use page numbers!)
>This is again intuitively clear, but
>the definition of I is unpleasant.
>
>I actually found the Riesz Representation Theorem much easier to follow.
>
>>
>> ************************
>>
>> David C. Ullrich
>
************************
David C. Ullrich
.
- References:
- obvious fact about volume
- From: NotP
- Re: obvious fact about volume
- From: NotP
- Re: obvious fact about volume
- From: David C . Ullrich
- obvious fact about volume
- Prev by Date: Re: 3rd Experiment |OR| Infinity
- Next by Date: Re: Lebesgue Integral Convergence
- Previous by thread: Re: obvious fact about volume
- Next by thread: Re: obvious fact about volume
- Index(es):
Relevant Pages
|
