Thread: Jordan Regions

1. Jordan Regions

Prove that all balls in R^n are Jordan regions.

For rectangles it is easy enough, but I can't seem to find a way of proving it for balls. Thanks for the help in advance!

2. Originally Posted by h2osprey
Prove that all balls in R^n are Jordan regions.

For rectangles it is easy enough, but I can't seem to find a way of proving it for balls. Thanks for the help in advance!
Do you mean that they're Jordan measurable sets?

3. Hmm I'm not sure about the terminology, but basically a Jordan set is one that's bounded, and its boundary has volume zero.

4. Originally Posted by h2osprey
Hmm I'm not sure about the terminology, but basically a Jordan set is one that's bounded, and its boundary has volume zero.
Yes they're the same, and you actually have the definition that's useful in this case!

Remember that $\mathbb{Q} ^n$ is countable and ennumerate it by $(x_n)_{n\in \mathbb{N} }$. Given $a>0$ take cover the sphere (boundary of the ball) with sets of the form $\{ B_{r_{x_n}}(x_n) : r_{x_n} < \frac{a}{2^n} \}=A$ then $A$ certainly covers your sphere and since this last one is compact it can be covered by a finite number of these, say (after possibly renaming them) $B=(B_{r_{x_i}}(x_i))_{i=1}^k$ then $vol(sphere) \leq vol(\cup B ) \leq \sum_{i=1}^{\infty } \frac{a}{2^i} =a$ so the volume of the sphere is bounded by an arbitrarily small number.

5. Originally Posted by Jose27
Yes they're the same, and you actually have the definition that's useful in this case!

Remember that $\mathbb{Q} ^n$ is countable and ennumerate it by $(x_n)_{n\in \mathbb{N} }$. Given $a>0$ take cover the sphere (boundary of the ball) with sets of the form $\{ B_{r_{x_n}}(x_n) : r_{x_n} < \frac{a}{2^n} \}=A$ then $A$ certainly covers your sphere and since this last one is compact it can be covered by a finite number of these, say (after possibly renaming them) $B=(B_{r_{x_i}}(x_i))_{i=1}^k$ then $vol(sphere) \leq vol(\cup B ) \leq \sum_{i=1}^{\infty } \frac{a}{2^i} =a$ so the volume of the sphere is bounded by an arbitrarily small number.
Thanks, I understand almost all of it, except the last part -

$vol(\cup B ) \leq \sum_{i=1}^{\infty } \frac{a}{2^i} =a$

Why is this inequality true? If we were trying to contain the spheres in cubes, should it not be $\sum_{i=1}^{\infty } (\frac{a}{2^i})^n$ instead?

jordan region math

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