Integrability and Lebesgue

[philosophking]

My analysis professor, a few weeks ago, when we were talking about integrability, introduced the concept of Lebesgue measure zero. He put up a theorem stating that the set of discontinuities of a function are of Lebesgue measure zero if and only if the function is integrable. This is, of course, after defining lebesgue measure zero.

I've tried to google for this and haven't found it. Is it normally introduced in an introductory real analysis class? Where exactly does Lebesgue measure come about from? References to books, etc. would be appreciated.

[Astronuc]

Have you tried Mathworld yet?

http://mathworld.wolfram.com/LebesgueMeasure.html

which references

Kestelman, H. "Lebesgue Measure." Ch. 3 in Modern Theories of Integration, 2nd rev. ed. New York: Dover, pp. 67-91, 1960.

[mathwonk]

my analysis prof introduced lebesgue integration this way:

In riemann integration we approximate the integral by dividing up the x interval into equal subintervals. then we take one point in each subinterval and evaluate the function there and multiply by the length of the subinterval. add these up and approximate the integral

this is obviously a stupoid way to approximate the integral since there is no reaon for the function not to have wildly different values on the subinterval even if rather small. e.g. if the function is discontinuous at apoint, then it may have wildly diferent values on every interval containing that point no matter how small. this amkes it ahrd to intregrate discontinuous functions by this method.

a much better way, if you think,about how you average a set of scores e,g,, is you ask how many people made a certain score and yopu multiply that score by the number of epople making that score.

so the good way is to divide up th y axis. i.e. sudivide the ya xis into various possible values and then ask how at many points the function ahd each range of values.

then of course you are faced with the challenge of measuring the size of the inverse image of an intervalk unders ome function and that can be a very strange loking set. lebesgue undertook to emasure the size of such sets.


then you start by trying how to cover the set with a sequence of intervals, and asking how short you can make the sum of the lengths of those intervals. that the "outer measure" of the set. you define the inner measure analogously and say the set ahs measure if those two numbers are the same.