Sets of measure zero

Despite common smouldering discontent about the concept of a measure zero, it is a very interesting and useful concept. I have been recently talking to a student and was surprised that the person hated the concept of measure zero, calling it completely useless. Hence I decided to clarify some things.

Consider a very easy case in \mathbb{R}^1. Recall that Riemann function

\mathcal{R} (x) := \frac{1}{n}, x \in \mathbb{Q} ~and~ x = \frac{m}{n}~ is~ in~ lowest~ terms,

\mathcal{R}(x) := 0, x \in \mathbb{R} \backslash \mathbb{Q},

on, say, the interval 0 to 1.

Recall how hard it was to prove that this function is Riemann integrable? We had to take some \epsilon > 0 and show that the limit of the function is everywhere 0 and… all different sorts of things.

Now consider a situation where we know the notion of a set with measure zero. That is, we can state the following

Definition: A set E \subset \mathbb{R} has or is of measure zero, if \forall \epsilon > 0, ~ \exists a cover of the set E by an a (finite or not) countable system \{I_i\} of intervals, the sum of lengths of which \sum\limits_{i=1}^\infty | I_i | \leq \varepsilon.

Then we can easily prove the fact that a set that consists of only the rational number between a and b is, in fact, a set of measure zero. Then use recall the Lebesgue criterion for Riemann integrability, which states

Theorem (Lebesgue Criterion for Riemann Integrability): A function defined on a closed interval is Riemann integrable, if and only if it is bounded on this interval and the set of points, where the function is discontinuous, is of measure zero.

We are now ready to solve the problem. We see that the Riemann function \mathcal{R}(x) is discontinuous at every rational point x \in \mathbb{Q}. Then the set of points, where the function is discontinuous, contains only the rational points and hence it is of measure zero. By the Lebesgue criterion for Riemann integrability, the Riemann function \mathcal{R}(x) is integrable on 0 to 1.

Easy? I thought so.

This entry was posted on Tuesday, January 15th, 2008 at 12:36 and is filed under Mathematics. You can follow any responses to this entry through the RSS 2.0 feed. You can leave a response, or trackback from your own site.

3 Responses to “Sets of measure zero”

  1. zerihun Says:
    Friday, August 7, 2009 at 1:58 | Reply

    you have gave me useful information about measure zero set defintion.Thank you . i think you will be my friend because now i am learning MEd(master of education) in mathematics

  2. zerihun Says:
    Friday, August 7, 2009 at 2:00 | Reply

    Thank you

  3. Nikita Says:
    Sunday, December 13, 2009 at 23:06 | Reply

    Zerihun,

    Apologies for such a late response.

    I am glad, though, that a portion of the things I put up here come in handy to some people, and twice as happy for the fact that there are those, like yourself, who devote their lives to educating others. This is certainly the greatest and most important element of any scientist’s social duty, which is sadly often forgotten by many.

    Please, teach us teach. And do so despite the so many mean things we may say back — some of us still listen ;)

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