A description/proof of that area on Euclidean metric space can be measured using only hypersquares, instead of hyperrectangles
Topics
About: metric space
The table of contents of this article
Starting Context
Target Context
- The reader will have a description and a proof of the proposition that any area on any \(\mathbb{R}^n\) Euclidean metric space can be measured using only hypersquares, instead of hyperrectangles.
Orientation
There is a list of definitions discussed so far in this site.
There is a list of propositions discussed so far in this site.
Main Body
1: Description
For any area, \(a\), on any \(\mathbb{R}^n\) Euclidean metric space, \(a\) can be measured using only hypersquares.
2: Proof
By the usual definition of area by the measure theory, for any real number, \(\epsilon \gt 0\), there are some countable number disjoint half-open intervals (hyperrectangles) that cover the area, \(\{R_i\}\), such that \(m (R_i) - a \lt \epsilon\) where \(m (\bullet)\) means the measure of the argument.
By the proposition that the area of any hyperrectangle can be approximated by the area of covering finite number hypersquares to any precision, a covering finite number hypersquares, \(\{S_{i_j}\}\), can be chosen such that \(\sum _j m (S_{i_j}) - m (R_i) \lt \frac{1}{k} m (R_i)\) where \(k\) is any natural number.
Then, \(\sum _i (\sum _j m (S_{i_j})) - a \lt \sum _i (m (R_i) + \frac{1}{k} m (R_i)) - a = \sum _i \frac{k + 1}{k} m (R_i) - a = \frac{k + 1}{k} \sum _i m (R_i) - a \lt \frac{k + 1}{k} (a + \epsilon) - a\). We want to choose \(k\) and \(\epsilon\) such that for any \(\epsilon'\), \(\frac{k + 1}{k} (a + \epsilon) - a \leq \epsilon'\). \(\frac{1}{k} a + \frac{k + 1}{k} \epsilon \leq \epsilon'\); we choose \(\frac{1}{k} a \leq \frac{\epsilon'}{2}\) and \(\frac{k + 1}{k} \epsilon \leq \frac{\epsilon'}{2}\), which is possible by \(k \geq \frac{2a}{\epsilon'}\) and \(\epsilon \leq \frac{k}{k + 1}\frac{\epsilon'}{2}\).
The hypersquares are countable, because we can count them, for example, by the order of \(i + j\) and then \(i\) for \({S_{i_j}}\), like \(S_{1_1}, S_{1_2}, S_{2_1}, S_{1_3}, S_{2_2}, S_{3_1}, . . .\).
So, for any real number \(\epsilon' \gt 0\), there are some countable number hypersquares, \({S_{i_j}}\), such that \(\sum _i \sum _j S_{i_j} - a \lt \epsilon'\).
