Thread: Integral Question with Cantor Set

1. Integral Question with Cantor Set

Exercise 6 in Chapter 6 of Baby Rudin is giving me some difficulties..

Let P be the Cantor set. Let f be a bounded real function on [0,1] which is continuous at every point outside P. Prove that f is Riemann integrable on [0,1].(Hint: P can be covered by finitely many segments whose total length can be made as small as desired. Proceed as in Theorem 6.10.)

I understand that their total length will be (2/3)^n and therefore we can show that the total length is less than epsilon by taking n sufficiently large, but how do we cover each interval in the set with finitely many segments when it is infinite..? I mean it seems like the total damage will be negligible and won't affect the integral, but I'm confused about how to go about this. Any help would be greatly appreciated, thank you.

2. When you look at the iterative construction of P (where you remove middle thirds recursively), you will notice that at each step, the part you haven't removed yet (which of course contains P) consists of a collection of intervals, call it C_n. So partition [0,1] using one of those collections, and consider the upper and lower sums for that partition. There are two cases: an interval does not belong to C_n, in which case f is continuous there, or the interval does belong to C_n, in which case you can put a bound on the contribution of that interval to the integral, using boundedness of f and the fact that the lengths of those intervals becomes arbitrarily small.

3. Thanks

Wow, that was a quick response. I didn't think I would get some help so quickly, thank you very much I understand what I must do now.

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