## 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…

There is, as we might expect, a relationship between outer Lebesgue measure and Jordan content, some aspects of which we will flesh out now.

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 .

Hey John. Sorry I haven’t commented for a while. I was wondering, what is the (or a?) Lebesgue measure?

Thanks in advancement and happy holidays,

NS

Comment by notedscholar | December 13, 2009 |

I haven’t defined it, and I don’t really intend to. I was pretty clear on this point when I started talking about Jordan content.

Comment by John Armstrong | December 13, 2009 |

Oh I missed that part. Sorry about the trouble.

NS

Comment by notedscholar | December 13, 2009 |

sweet jesus, a blog where people talk abt math – recreationally?

Comment by none | December 13, 2009 |

Yes, and you know this already because I recognize your fake email address.

Comment by John Armstrong | December 13, 2009 |

oh snaps!!!

lol,

NS

Comment by notedscholar | December 21, 2009 |

[…] , we also have , and therefore have as […]

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[…] measurable, because this will happen if and only if the boundary has zero Jordan content, and thus zero outer Lebesgue measure. Since the collection of new discontinuities must be contained in this boundary, it will also have […]

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