- #1
jinsing
- 30
- 0
Homework Statement
If f is bounded on the measurable set S, and the measure of S is finite, and P,Q are partitions of S, then L(f,P) \leq U(f,Q)
Homework Equations
Lebesgue measurability/integrability, refinements
The Attempt at a Solution
Not sure if this is totally right, but:
Assume the hypotheses. Let P = {E1, E2,...,En} and Q={F1, F2,...,Fm}. Let R = {A_ji}, where [tex] E_j \cap F_i = \bigcup_{i=0}^{k_j} A_{ji}. [/tex] Then R is a refinement of P \cap Q. Since S has finite measure and is measurable on a bounded function f, then it follows that
[tex] m \mu(S) \leq L(f,P\cap Q) \leq L(f,R) \leq U(f,R) \leq U(f, P \cap Q) \leq M \mu(S). [/tex]
But if R is a refinement of P \cap Q, then clearly R is a refinement of P and R is a refinment of Q. Then it follows that L(f,P) \leq U(f,Q).
You might be able to tell, but I was sort of grasping for straws towards the end. Any useful hints?
Thanks!