I am having hard time understanding the definition of zero content. The following are the definitions of zero content in $\mathbb{R}$ and $\mathbb{R}^2.$

  1. A set $Z \subset \mathbb{R}$ is said to have zero content if $\forall \epsilon > 0$ there is a collection of intervals $I_1, \ldots, I_L$ such that
    (i) $Z \subset \bigcup_1^L I_l,$ and
    (ii) the sum of the lengths of the $I_l$ is less than $\epsilon.$
  2. A set $Z \subset \mathbb{R}^2$ is said to have zero content if $\forall \epsilon > 0$ there is a finite collection of rectangles $R_i$ such that
    (i) $Z \subset \bigcup_1^M R_i$ and
    (ii) the sum of areas of the $R_i$ is less than $\epsilon.$


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    $\begingroup$ What do you mean by "clarify the meaning"? What kind of answer are you looking for here? $\endgroup$ – Omnomnomnom Nov 20 '14 at 16:47
  • $\begingroup$ I am looking for an easier explanation of those definitions with examples of pictures if possible. $\endgroup$ – eChung00 Nov 20 '14 at 16:49
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    $\begingroup$ There are some pretty pictures on the relevant wikipedia page $\endgroup$ – Omnomnomnom Nov 20 '14 at 16:53

Update: the answer below is not correct, see comments.

Essentially, that means that Lebesgue measure of $Z$ (length in $\Bbb R$, area in $\Bbb R^2$ etc.) is zero. Measures are monotone, i.e. $\lambda(Z)\leq \lambda (\cup I_l)$ since $Z\subseteq \cup I_l$. Measures are additive, that is $\lambda (\cup I_l) = \sum \lambda(I_l) \leq \epsilon$. Hence, you get $\lambda (Z)\leq \epsilon$ for all $\epsilon>0$ and so $\lambda(Z) = 0$.

Measure theory is a bit abstract, and formal definition of $\lambda$ is rather technical. Yet, in early applications in analysis you often just need to know that something has measure of $0$, and you don't care much what is the measure whenever it is greater than $0$ - for example, you want boundaries of nice sets to have measure of $0$. For this reason, instead of giving the whole definition of measures, you simply define what does it mean to be of zero measure, or of zero content in your terminology.

  • $\begingroup$ You should clarify what you mean by "essentially". Note, for example, that neither $\Bbb Q$ nor $\Bbb Q \times \Bbb Q$ have content zero by this definition. $\endgroup$ – Omnomnomnom Nov 20 '14 at 16:50
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    $\begingroup$ It's not quite the same as zero Lebesgue measure, is it? It doesn't allow the cover to be infinite. So an unbounded set can't have zero content. $\endgroup$ – TonyK Nov 20 '14 at 16:51
  • $\begingroup$ That's what I thought too, but it looks like the concept in the OP wants a finite collection of intervals/boxes, so it doesn't get all of the Lebesgue null sets. $\endgroup$ – Henning Makholm Nov 20 '14 at 16:53
  • $\begingroup$ @Omnomnomnom: I am not sure whether $M,L$ are assumed to be finite. $\endgroup$ – Ilya Nov 20 '14 at 16:53
  • $\begingroup$ @HenningMakholm: perhaps, you are right. The link by Omnomnomnom made in the comment to OP looks more relevant. $\endgroup$ – Ilya Nov 20 '14 at 16:54

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