http://www.tau.ac.il/~tsirel/Courses...sis3/lect6.pdf
Hi,
i need help with the solution of problem 6h3 (c),on page 87. in the attached link. for p not smaller than 1,i think i know,but there must be a nice and compact solution.
Thanks in advance.
http://www.tau.ac.il/~tsirel/Courses...sis3/lect6.pdf
Hi,
i need help with the solution of problem 6h3 (c),on page 87. in the attached link. for p not smaller than 1,i think i know,but there must be a nice and compact solution.
Thanks in advance.
Are you allowed to use results from topology in your proof? Define $\displaystyle d: \mathbb{R}^n \times \mathbb{R}^n \to [0,\infty )$ by
$\displaystyle d(x,y) = \sup_{1\le i\le n}|x_i - y_i|$
The topology this generates is frequently called the "Box metric". It is a routine exercise to show that these two metrics are topologically equivalent. A very similar proof can be used to show that each $\displaystyle E_p, p>0$ is Jordan measurable.
Sorry, by two metrics, I meant the Euclidean metric and the Box metric are topologically equivalent. Those were the two metrics I meant. To show their topological equivalent, show that about every point of an open ball, there exists an open box centered at that point lying completely inside of the open ball. Then, show that about every point inside of an open box there exists an open ball centered at that point sitting inside of the box. (That is the topological equivalence proof). For Jordan measurability, you need to do something similar. Show that the $\displaystyle \sigma$-algebra generated by closed balls is equivalent to the $\displaystyle \sigma$-algebra generated by boxes. Since the set of Jordan measurable sets is a $\displaystyle \sigma$-algebra containing all boxes, it must contain the $\displaystyle \sigma$-algebra generated by closed balls, and the problem is solved.