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Consider a point in a line, it has length zero . Consider a line in a plane , it has area zero . Consider a Plane in $\mathbb R^3$ it has $0$ volume .

My question is , is there a measure theoretic way of extracting these ideas ? How can i link these ideas with Measure theory ?

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2 Answers 2

up vote 4 down vote accepted

Given $Y$ is a proper subspace of $X$ ($\dim Y < k$), then $m(Y) = 0$. It's intuitively clear, but it's not trivial!!

It is true for $R^k$, because the Lebesgue measure is outer-regular (ie: for every $A$ measurable, $m(A) = \inf\{m(E): A\subseteq E,\, E\text{ open}\}$.

$Y$ is a proper subspace of $X$, by linear transformation, you can move it to another subspace $E$ with the last coordinate is $0$. $E = \{(x_1,x_2,\ldots ,x_k): x_k =0\}$, and it is sufficient to show $m(E) = 0$. (" the sufficient part here " is actually not trivial: there is theorem in Rudin said that, if you have a linear transformation $T$, for any set $E$, $m(T(E)) = \lambda m(E)$, $\lambda$ is constant, and I need this theorem to use this linear transformation.)

The set of $S = \{(x_1,x_2,\ldots ,x_{k-1},0)\}$, with every $x_i$ rational, is countable, and can be listed as $y_1,y_2,y_3,\ldots$

For each point $y_i = (x_1,x_2,...x_{k-1},0)$, we put it inside a box $W_i$ $$W_i = (x_1-1/2,x_1+1/2) \times (x_2-1/2,x_2+1/2)\times \ldots\times (x_{k-1}-1/2,x_{k-1}+1/2) \times(-\epsilon/2^{i+1},\epsilon/2^{i+1})$$

The volume/measure of $W_i$ is $\epsilon/2^i$.

$$E = \bigcup W_i \implies m(E) \le \sum\limits_{i=1}^\infty m(W_i) = \epsilon \implies m(E) =0.$$

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by the way, this one is problem 2.16 in Rudin, if you want more awesome problems!! –  user1412 May 16 '12 at 11:00
    
For an implication, I recommend using "\Rightarrow" ($\Rightarrow$) or "\Longrightarrow" ($\Longrightarrow$) instead of "=>". –  Quinn Culver May 16 '12 at 13:21
    
@QuinnCulver, Thank you very much for your suggestion. I didn't know about \rightarrow before. I feel bad about => either, as it looks confused with <= or =. –  user1412 May 16 '12 at 13:43
    
Even better \implies $\implies$, \impliedby $\impliedby$ and \iff $\iff$. –  leo May 16 '12 at 16:36
    
thank you leo :D –  user1412 May 16 '12 at 20:25

For every $n\in\mathbb{N}$ you can consider the Lebesgue measure $m_{n}$ on $\mathbb{R}^{n}$: it assigns the intuitive volume as one imagines to a (measurable) subset of $\mathbb{R}^{n}$. In particular, in $\mathbb{R}^{n}$ every $n-1$ "dimensional" object has the corresponding $m_{n}$ Lebesgue measure of zero. E.g. a point has zero $m_{1}$-measure, a line has zero $m_{2}$-measure, a plane has zero $m_{3}$-measure, etc.

Examples of measures which are less coarse and take these "lower dimensional" objects into account very nicely are Hausdorff measures.

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