Let f be a (Lebesgue) integrable function and A be a measurable set.
Show that:
for some K
I know that since f is integrable we can say , and this feels like it should be a two line proof, but I can explicitly show how to get to the next step.
Let f be a (Lebesgue) integrable function and A be a measurable set.
Show that:
for some K
I know that since f is integrable we can say , and this feels like it should be a two line proof, but I can explicitly show how to get to the next step.
Disregard the following... (facepalm)
Let s be a simple function, so . Call such a function "good". We aim to show the integral of any good function is bounded by so is too.
Since |f| is integrable, the set is bounded above and we can chuse a simple function whose integral is close to the supremum. Then is simple and greater than every function in E for large k (oopsthis part was wrong; thanks Opalg). Suppose s is a good function. Then s is in E, so t is greater than it, and it follows that t1_A is greater than it. So for all good s and it follows that for all good s so that . Now since t is simple, , so and we are done.
(Note that I've abused notation by writing E for both the set of all simple functions between 0 and |f|, and the integrals of such functions. I hope its not confusing.)