1-1 curve with lebesque measure greater than 0
analysisman
is is probably some sort of modification of a space filling curve
analysisman
Jon Slaughter
"analysisman" <partypo...@yahoo.com> wrote in message
news:7413286.11145641196...@nitrogen.mathforum.org...
> can anyone think of a 1-1 curve with lebesque measure greater than 0?
> is is probably some sort of modification of a space filling curve
What is the lebesque measure of x*(D(x)+1) where D(x) is the Dirichlet
function?
Jon
David C. Ullrich
I don't know what the Dirichlet function is, but if the curve you
have in mind is the graph of a function, as sounds likely, then
it's clear that it has measure zero (think about Fubini's theorem...)
>Jon
>
************************
David C. Ullrich
G. A. Edgar
<7413286.11145641196...@nitrogen.mathforum.org>,
analysisman <partypo...@yahoo.com> wrote:
> can anyone think of a 1-1 curve with lebesque measure greater than 0?
> is is probably some sort of modification of a space filling curve
Lebesgue 1903, Osgood 1903...
See Mandelbrot, THE FRACTAL GEOMETRY OF NATURE, chapter 15.
p. 148: curves with positive area
--
G. A. Edgar http://www.math.ohio-state.edu/~edgar/
Daniel Grubb
>can anyone think of a 1-1 curve with lebesque measure greater than 0?
>is is probably some sort of modification of a space filling curve
Yes, there are such. One way to construct one is to consider
two 'fat' Cantor sets in [0,1], i.e. Cantor sets of strictly
positive measure. In [0,1]^2, define your curve as a limit of
curves that go through all the 'corners' obtained from endpoints
of each stage of the construction of the Cantor sets. With care,
you obtain a 1-1 curve going through all the points in the cross
product of the Cantor sets, so the curve has measure more than 0.
--Dan Grubb
David C. Ullrich
Well such things certainly exist.
For example you could start with a curve with a lot of vertical
zig-zags. Then at the next stage each vertical segment is replaced
by a curve with much shorter and much more densely packed horizonal
zigzags, at the next stage each of those horizontal segments is
replaced by more vertical zigzags, etc. If you do that right the
limit should be a 1-1 curve with positive area.
************************
David C. Ullrich
Zdislav V. Kovarik
I wonder if it does not clash with the fact that the planar Lebesgue
measure of the graph of a Lebesgue measurable function is zero. At each
step above, if you keep the curve 1-1, you keep the measure at 0, since
Lebesgue measure is sigma-additive. Or did I miss something?
Cheers, ZVK(Slavek).
analysisman
Thanks, this sounds like the most promising solution for me to explore.
David C. Ullrich
<kov...@mcmaster.ca> wrote:
>
>
>On Wed, 27 Apr 2005, David C. Ullrich wrote:
>
>> On Tue, 26 Apr 2005 21:08:09 EDT, analysisman
>> <partypo...@yahoo.com> wrote:
>>
>> >can anyone think of a 1-1 curve with lebesque measure greater than 0?
>> >is is probably some sort of modification of a space filling curve
>>
>> Well such things certainly exist.
>>
>> For example you could start with a curve with a lot of vertical
>> zig-zags. Then at the next stage each vertical segment is replaced
>> by a curve with much shorter and much more densely packed horizonal
>> zigzags, at the next stage each of those horizontal segments is
>> replaced by more vertical zigzags, etc. If you do that right the
>> limit should be a 1-1 curve with positive area.
>>
>> ************************
>>
>> David C. Ullrich
>
>I wonder if it does not clash with the fact that the planar Lebesgue
>measure of the graph of a Lebesgue measurable function is zero.
The approximating curves are not graphs of functions, not that that
matters since yes, they do have zero area.
>At each
>step above, if you keep the curve 1-1, you keep the measure at 0, since
>Lebesgue measure is sigma-additive. Or did I miss something?
You missed something. Not that it's really fair to put it that way,
since all I gave was a vague sketch, but your objections are no
problem.
Yes, at each stage of the construction we have a curve with area
0, but the curve we finally construct is the _limit_ of those curves,
not the union.
I'm not going to try to describe things more precisely and prove that
the limit has positive area in a usenet post. Instead I'll just point
out that the standard constructions of space-filling curves construct
them as limits of curves with area 0 - if your objection were valid it
would apply to those space-filling curves as well, ultimately leading
to a proof that the plane has Lebesgue measure 0.
>Cheers, ZVK(Slavek).
************************
David C. Ullrich
Dave L. Renfro
The best references I know of for someone who is
just beginning to look at these things, and who
wants a fairly thorough treatment with all the
details included, are:
Karl Stromberg and Shio Jenn Tseng, "Simple plane arcs
of positive area", Expositiones Mathematicae 12 (1994),
31-52. [MR 95b:28004; Zbl 797.28003]
http://www.emis.de/cgi-bin/MATH-item?0797.28003
Hans Sagan, SPACE-FILLING CURVES, Springer-Verlag, 1994.
[See Chapter 8 (pp. 131-143).] [MR 95h:00001; Zbl 806.01019]
http://www.emis.de/cgi-bin/MATH-item?0806.01019
Reviewed by John Holbrook in The Mathematical
Intelligencer 19(1) (Winter 1997), 69-71.
Incidentally (this is for others who might be reading),
when Osgood wrote his paper (submitted in Nov. 1902,
published in Jan. 1903), Lebesgue measure was not yet
generally known, and thus Osgood used outer Peano-Jordan
content in R^2. However, since the curve he constructed
is a compact subset of R^2 (it's a continuous image of
[0,1]), his curve actually has positive Lebesgue measure.
Note that this is a stronger result, since having positive
Lebesgue measure implies positive outer Peano-Jordan content,
but not conversely (even for bounded sets). For example,
the rationals in R (and pairs of rationals in R^2) have
full outer Peano-Jordan content and yet have Legesgue
measure zero.
This is an example of how a Borel condition can influence
the size and/or density of a set. This idea arises quite
often and it can be a useful tool, but I don't believe
I've ever seen this technique discussed in a textbook.
Here are some other examples of this phenomena in R^n:
* An uncountable Borel set has cardinality c.
* A dense G_delta set is co-meager, and hence co-meager
in every open ball, and hence c-dense (and more)
in R^n.
* A G_delta meager set is nowhere dense.
* A closed Lebesgue measure zero set is nowhere dense.
* An F_sigma Lebesgue measure zero set is meager.
* A G_delta countable set is scattered.
(Scattered sets are not only nowhere dense in R^n,
but they are even nowhere dense relative to every
nonempty perfect subset of R^n.)
* A Borel set with cardinality < c is countable.
Dave L. Renfro
Lee Rudolph
...
>Here are some other examples of this phenomena in R^n:
>
>* An uncountable Borel set has cardinality c.
...
>* A Borel set with cardinality < c is countable.
For appropriate values of "other".
Lee Rudolph
Dave L. Renfro
>> Here are some other examples of this phenomena in R^n:
>>
>> * An uncountable Borel set has cardinality c.
...
>> * A Borel set with cardinality < c is countable.
>
> For appropriate values of "other".
I'm guessing this is a nit-pick correction, but I can't
figure out what you meant. What did I leave out?
(By asking this I'll probably notice what it is right
after I submit this post, in the same way that carrying
an umbrella on a cloudy day will ensure that it's not going
to rain.)
Dave L. Renfro
Lee Rudolph
Well, having already demonstrated that today I'm working hard to
stay above water, I will merely point out (without claiming that
I follow my own argument, or that anyone else can or should) that
it seems to me that the second starred statement is a rather
immediate consequence of the first: for, given the first
starred statement, and a Borel set X that is *not* countable,
it follows that the cardinality of X is *not* < c (for it is
c). Yes?
Lee Rudolph
Dave L. Renfro
>>>> This is an example of how a Borel condition can influence
>>>> the size and/or density of a set. This idea arises quite
>>>> often and it can be a useful tool, but I don't believe
>>>> I've ever seen this technique discussed in a textbook.
>>>> Here are some other examples of this phenomena in R^n:
>>>>
>>>> * An uncountable Borel set has cardinality c.
>>>>
>>>> * A dense G_delta set is co-meager, and hence co-meager
>>>> in every open ball, and hence c-dense (and more)
>>>> in R^n.
>>>>
>>>> * A G_delta meager set is nowhere dense.
>>>>
>>>> * A closed Lebesgue measure zero set is nowhere dense.
>>>>
>>>> * An F_sigma Lebesgue measure zero set is meager.
>>>>
>>>> * A G_delta countable set is scattered.
>>>> (Scattered sets are not only nowhere dense in R^n,
>>>> but they are even nowhere dense relative to every
>>>> nonempty perfect subset of R^n.)
>>>>
>>>> * A Borel set with cardinality < c is countable.
Lee Rudolph wrote:
>>> For appropriate values of "other".
Dave L. Renfro wrote:
>> I'm guessing this is a nit-pick correction, but I can't
>> figure out what you meant. What did I leave out?
Lee Rudolph wrote:
> Well, having already demonstrated that today I'm working
> hard to stay above water, I will merely point out (without
> claiming that I follow my own argument, or that anyone
> else can or should) that it seems to me that the second
> starred statement is a rather immediate consequence of
> the first: for, given the first starred statement, and
> a Borel set X that is *not* countable, it follows that
> the cardinality of X is *not* < c (for it is c). Yes?
Oh, O-K. My intent was to simply list some examples
that can arise in practice without worrying about their
independence.
The statements you're talking about (for anyone else who
might be reading) concerned my 1'st and 7'th starred
statements, which became "first" and "second" when you
repeated them without the other statements.
Note that my 3'rd statement also follows easily from my
2'nd, my 5'th statement is immediate from my 4'th, and
you can get my 6'th statement from my 3'rd along with
the Baire category theorem (where the ambient space
is a nonempty perfect subset of R^n). (If I got any
of these arguments backwards or mixed up, I'll leave it
as an exercise for sci.math readers to fix.)
I grouped them roughly according to how large and/or dense
a Borel condition forces the set to be, by the way. The
first two allow us to interpolate to a larger size and/or
density, while the last five allow us to interpolate to
a smaller size and/or density.
Dave L. Renfro