Outer Lebesgue Measure and Content
Before I begin, I’d like to mention something in passing about Lebesgue measure. It’s pronounced “luh-bayg”. The “e” is a long “a”, the “s” is completely silent, and the “gue” is like in “analogue”. Moving on…
First off, if is a bounded subset of -dimensional Euclidean space, then we have . Indeed, if it’s bounded, then we can put it into a box , and choose a partition of this box. List out the -dimensional intervals of which contain points of as for . Then by definition we have
Now given an , define the open -dimensional intervals . These form a Lebesgue covering of for which
Thus , and passing to the infimum we find . Since is arbitrary, we have .
Secondly, if is bounded, and is a compact subset, then . Start by putting into a box , and take some . We can find a Lebesgue covering of so that , and this will also cover . Since is compact, we can pick out a finite collection of open intervals which still manage to cover . Finally, we can choose a partition of so that the corners of each interval in are partition points. Given all of these choices, we find
And since is arbitrary we conclude .
Finally, putting these two results together we can see that if is a compact set, then .