I was told this was a rather easy question, but I'm having troubles sorting it all out....
Given that is an integrable function on , prove
1) For any ,
2) For any ,
The proof is said to be simplest by starting with the direct definitions of integrability.
This is what I've got so far...
For 1)
taking the definition of an integral, we know that the integral is equal to the supremum of the lower sums and the infimum of the upper sums:
sup inf
taking this a step farther, we can put this into summation notation:
sup
Then it's as easy as seeing that since has become , we now have
sup
then
sup
then
sup
then
sup
so it's obvious where the comes from, it's the and of the integral that i can't quite explain
So I've finished the proof! YAY! But...I've come up with a slightly different answer. Have i gone wrong somewhere? Is the question wrong? Any comments will help, thank you!
pf//
taking the definition of an integral, we know that the integral is equal to the supremum of the lower sums and the infimum of the upper sums:
sup inf
taking this a step farther, we can put this into summation notation:
sup
Now we see that since becomes , we essentially have a mapping and our image is
sup
from here we simplify and bring our constant completely out
sup
sup
sup
Equating this to the left side of the equation after mapping, we see that
As for the change in the limits of integration, we first note that our mapping of has shifted our interval
to obtain our origional interval, we must compose our limits of integration with the inverse mapping function so we get
putting this all together, we see that
Q.E.D (sort of)
So.....why am i getting when i "should" be getting just ???