Re: line segment
- From: The World Wide Wade <aderamey.addw@xxxxxxxxxxx>
- Date: Wed, 26 Sep 2007 18:30:31 -0700
In article <fdb49p$rpt$1@xxxxxxxxxxxxxxxxxx>,
magidin@xxxxxxxxxxxxxxxxx (Arturo Magidin) wrote:
In article <1190728769.596021.46670@xxxxxxxxxxxxxxxxxxxxxxxxxxx>,
Jane <jane.oper@xxxxxxxxx> wrote:
Hello
Definition:
Let A be a subset of R^2. A has content zero if for all epsilon>0
exists
Q_1,Q_2,...,Q_n rectangles such that A is a subset of Q_i (i=1...n)
and sum(i=1...n) Area(Q_i)< epsilon.
Prove that a line segment has content zero.
How to solve this?
By enclosing the line segment in lots of really, really, thing
rectangles.
But "rectangle" here probably means "rectangle with sides parallel to
the axes", so "thin" may not be the way to go for the line segment
[(0,0), (1,1)] for example. That might be best covered by n squares of
side length 1/n. (In general, if f : [a,b] -> R is continuous (hence
uniformly continuous), the graph of f is contained in U I_m x f(I_m),
where the I_m partition [a,b], and f(I_m) will be small if the I_m are
small. Hence the graph of f has content 0.)
.
- References:
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- From: Jane
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- From: Arturo Magidin
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