The discussion revolves around proving that the set N(f) = {x : f(x)≠ 0} is σ-finite when f is integrable. Participants are exploring the implications of integrability on the nature of this set.
The discussion is ongoing, with participants providing insights and clarifications regarding the definition of σ-finiteness and its application to the set N(f). There is an exploration of the relationship between the measure of N(f) and countable unions of finite measure sets.
Participants are considering specific examples and definitions, such as the measure of sets A(f,n) and their implications for the proof. There is an acknowledgment of the need for clarity in the definitions being used.
manuel huant said:Homework Statement
if f is integrable, then the set N(f) = {x : f(x)≠ 0} is \sigma-finite
Homework Equations
i am stucked in this proof , somebody help me pleaseThe Attempt at a Solution
if f is simple the it seems the set is finite since otherwise the the integral won't exist but how can it be extended to f is integrable?
manuel huant said:sorry i didn't put it right , it seems N(f) \sigma-finite means the measure of N(f) is a countable union of finite measure sets
u(N(f))= \underbrace{\cup}_{n}u(N(fn)) which u(N(fn)) <\infty