I would like to know how to do this problem before my test tomorrow and I'm not really sure how to do it. Any guidance would be appreciated.
Let be the Cantor set. Let be determined by
Show that is Riemann integrable on and . [That is the "length of the Cantor set is 0.]
I want to approach it like this but I'm not sure if it is correct:
We want to show that is integrable we need to show that for some .
We can make when . So i think all we have to do is to show that . However, I'm not sure how to do this.
Ok, here is what I want to say but I'm not sure how to write it correctly of if its even in the right direction...
If we let be a lower step function of , then we see that . Then if we choose some upper step function, , such that where then for any interval contained in , we can find an element, that is contained in the Cantor set and is infinitely close to another element that is not contained in the Cantor set. Thus, we have . Thus, .
Thus, which is less then any . Thus, .
I know that is not exactly the direction that you were leading me to but I think it is similar.
I think that it would be better to show that the upper sum and lower sum are equal to each other. Recall that
and
where refers to some partition of . We say that is Riemann-integrable if .
Now I propose that we consider a sequence of partitions based on the cuts performed in the construction of the Cantor set. Let (corresponding to the removal of the middle third of the interval). Then, . Let (again, corresponding to the middle thirds of the remaining intervals). Then, . Perform this same procedure to get . Now taking , we have and . This shows that
and
or in other words,
But at the same time, from the definition of the upper and lower sums,
Therefore , so this function is Riemann-integrable with integral value 0.