Re: Idiosyncratic (or idiotic?) notation for lim inf, etc.

On Sun, 17 Apr 2005 14:19:36 -0500, David C. Ullrich
<ullrich@xxxxxxxxxxxxxxxx> wrote:

>On Sun, 17 Apr 2005 19:06:30 +0100, Angus Rodgers
><angus_prune@xxxxxxxxxxx> wrote:
>
>>Having blurted out in passing in another thread that I was
>>struggling to understand how to interrelate several forms
>>of Fatou's Lemma [...]

Well, someone responded to that, and things snowballed!
I didn't start with any intention of posting any of this
stuff.

>>That may already be a bad idea, for all I know - does anyone
>>else but JSH use sci.math to `think aloud'?
>
>I don't see why it's a _bad_ idea, but I'm not sure I see
>the point to it, unless you've come up with new and
>different proofs. The standard proofs are not all
>that hard, and they're easy to find in lots of books.

One point there is to it, I think, is that there doesn't
seem to be a single set of "standard proofs". I found
it difficult to organise the material in my mind, and,
having done so, I thought it /might/ be a good idea to
post the results in case anyone else was interested in
the same topic at the same time.

(In a deeper sense, of course, there's nothing new here.
I'm perfectly well aware of that.)

More specifically, it wasn't obvious to me when I started
that there was a single form [FL] of Fatou's Lemma such that
(in the course of the Daniell-style development, at least)
one can argue [Monotone Convergence Theorem] => [FL] =>
[everything else, in a few steps, without technicalities].

If sci.math turns out not to be a suitable place for this,
I could (as I eventually will, anyway) discuss the topics
with other students on the same course (it's distance
learning, with some informal e-mail communication), but I
thought a test drive here first would be wise, in case I
only spread confusion!

Possibly I've got that arse-backwards, and I'm only spreading
confusion (and/or boredom) here, whereas other students on
the same course would instantly see the point. If so, sorry.

(Oh, and my tutor generally only checks his e-mail once a
week - otherwise I would probably have chanced dumping all
this on him, and risking his displeasure instead of yours.)

>> - but before I
>>even get started on it, I want to check whether it might be
>>an even worse idea to use in public the notation that I have
>>been using in my own notes.
>
>That strikes me as a bad idea.

For any particular reason?

>>[...] Here I'll just mention that in [Weir], no
>>use is made of the extended real number system; and I don't
>>know if other books define the integral in such a way that
>>its value can sometimes be +oo or -oo. Perhaps there's
>>some `obvious' reason why this can't be done?
>
>No, it's very standard to say that every non-negative
>measurable function has an integral, possibly infinite;
>if f >= 0 then one often just defines the integral to
>be sup int g, for simple functions g with 0 <= g <= f.

OK, thanks. No doubt I'll be able to swot this up after
finishing Weir's book (unless it's in the later chapters
- probably is, I should imagine).

>Uh, this is the end of the post and you haven't made any
>mathematical assertions yet? It's really not clear what
>the point to this is. Oh, I see another thread where you
>do nothing but state the Monotone Convergence Theorem.
>
>I think that maybe if you're going to do this for whatever
>reason then at least making it all one thread would be
>better.

Originally I started to write it as one article, then
I thought: (a) people don't like long articles; and (b)
if my notation is crap, then I'd better find out before
imposing it on people - the rest of the long article
would then be pretty pointless.

>I mean the point to this thread is to set up the
>notation you intend to use in other threads, so those
>other threads are going to be assuming that the reader
>has read all your previous posts, in various different
>threads. Seems presumptuous or something.

You're not much of a one for giving people the benefit
of the doubt, are you? Whether misguidedly or not, I
was trying to be considerate - not the opposite.

By the way, why do you quote everything, even the bits
that you aren't replying to?

--
Angus Rodgers
(angus_prune@ eats spam; reply to angusrod@)
Contains mild peril
.

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