It's an excerpt from Rudin's book. I can't understand the following moments:
1) Why he considers continuous function with compact support? Why compactness is so important?
2) Why equation (3) has the meaning? Why $f$ is zero on the complement of $I^k$?
3) Why integral in (3) is independent of the choice $I^k$? It's not obvious to me.
Can anyone give an answer to my above questions? I would be very grateful.