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I stumbled upon these examples that weren't shown in the lecture:

Relevant Definitions:

  1. If $S\subset \mathbf{R}^{n}$ is measurable, and $Q$ is a cuboid in $\mathbf{R}^{n}$ which contains $S$ then $\chi_{S}: Q \rightarrow \mathbf{R}$ describes the characteristic function of $S$ on $Q$, defined by: $\chi_{S}(x)=\begin{cases} 1&\mbox{ if }x\in S,\\\ 0& \mbox{ else.}\end{cases}$ Then the set $S$ is a $n$-dimensional volume given by: $v_{n}(S) := \int_{S} 1d^{n}x = \int_{Q}\chi_{S}$.

  2. A cuboid in $\mathbf{R}^{n}$ is the product of $n$ closed intervals of the form: $Q= [a_{1},b_{1}]\times [a_{2},b_{2}]\times \cdots \times [a_{n},b_{n}] = \{(x_{1},\cdots,x_{n})\in \mathbf{R}^{n} | a_{j}\le x_{j} \le b_{j}\}$, where $a_{j},b_{j} \in \mathbf{R}$. The volume of the $n$ dimensional cuboid is given by : $v_{n}(Q):= \prod_{j=1}^{n} ( b_{j}-a_{j})$.

Examples:

  • If $B\subset \mathbf{R}^{3}$ is a measurable bounded set (which can even be fully contained in a plane E), then $v_{2}(\lambda B) = \lambda ^{2} v_{2}(B)$.

  • If $p \in \mathbf{R}^{3}$ is a point with distance $h>0$ of the plane E, then the cone K with basal area B and apex p is the convex Hülle $K=\{(1-s)x+sp | x \in B , 0 \le s \le 1 \} \subset \mathbf{R}^{3}$ and $v_{3}(K) = \frac{h}{3} v_{2}(B)$.

Showing the examples:

1.Geometrically, it makes sense because the area is given by the product of two lengths, so if you multiply each length by a factor, then that is the same as multiplying the area by the factor squared. I tried showing it formally like this:

$$v_{2}(\lambda B) = \int_{\lambda B} 1 d^{2}x = \int_{[a_{1},b_{1}]\times [[a_{2},b_{2}]} \chi _{\lambda B} .$$

But I am stuck there.

  1. Again, I understand geometrically why this is the case, but how to show this using measure theory.

Any help is greatly appreciated.

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For the first question, I suggest you to prove the result for cuboids, then for open sets, then for bounded $G_\delta$ sets, and finally for bounded measurable sets. –  leo Dec 15 '11 at 19:15

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