I know the definition, But I'm looking for an explanation.. anyone?

Printable View

- Apr 12th 2011, 06:41 AMcalculuskid1What is zero content?
I know the definition, But I'm looking for an explanation.. anyone?

- Apr 12th 2011, 06:47 AMPlato
- Apr 12th 2011, 07:21 AMcalculuskid1
I have:

for every epsilon>0 there is a finite collection of intervals I1...IL st:

A) Z⊂U IL,and

B) The sum of the lengths of the I l's is less than epsilon.

But how can the sum of the intervals be less than epsilon, does it just mean that zero content are intervals that are arbitrary small..? - Apr 12th 2011, 07:53 AMOpalg
For a start, any finite set of points has zero content. Given a set with n points, you can surround each of them by an interval of length $\displaystyle \varepsilon/n$.

For an example of an infinite set with zero content, let $\displaystyle S = \{1/n:n=1,2,3,\ldots\}$. Given $\displaystyle \varepsilon>0$, the interval $\displaystyle (0,\varepsilon/2)$ contains all but finitely many elements of S, and you can use the remaining $\displaystyle \varepsilon/2$ of length to put little intervals around those finitely many points. - Apr 12th 2011, 08:01 AMPlato
Suppose that $\displaystyle \varepsilon > 0$ the open interval $\displaystyle O_n = \left( {a - \frac{\varepsilon }{{2^{n + 1} }},a + \frac{\varepsilon }{{2^{n + 1} }}} \right)$ has length $\displaystyle \ell (O_n ) = \frac{\varepsilon }{{2^n }}$.

The sum of all those is just $\displaystyle \sum\limits_{n = 1}^\infty {\ell (O_n )} = \varepsilon $.

Thus $\displaystyle \{a\}$ has content zero.

Now that is a simple-minded example that has a natural extension to larger sets.

Essentially the set has ‘zero area’. Think about any open interval say $\displaystyle [a,b]$.

If $\displaystyle \varepsilon = \frac{{b - a}}{2} > 0$ it would be impossible to cover $\displaystyle [a,b]$ with set whose total length is less that $\displaystyle \varepsilon $.

Edit: I did not see reply #4 before posting, sorry. - Apr 13th 2011, 10:59 AMDeveno
content is supposed to correspond intuitively to length (in 1 dimension), area (in 2 dimensions) or volume (in higher dimensions).

i say "supposed" because it turns out that historically, this has been a hard notion to pin down. the problem isn't with "normal" things like lines, triangles, circles, squares, etc. it has to do with the fact that there are lots and lots of kinds of sets, and some of them are pretty weird.

consider this set: {x in [0,1] : x is rational}. what is the "length" of this set? how would we even begin to measure it? it turns out that different notions of measurement, give different answers.