# Proof of a set is sigma finite

1. Jan 23, 2013

### manuel huant

1. The problem statement, all variables and given/known data

if f is integrable, then the set N(f) = {x : f(x)≠ 0} is $\sigma$-finite

2. Relevant equations
i am stucked in this proof , somebody help me please

3. The 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?

2. Jan 23, 2013

### Dick

What's the definition of N(f) being sigma finite? N(f) doesn't have to be finite. f(x)=1/x^2 is integrable on [1,infinity). [1,infinity) isn't finite.

Last edited: Jan 23, 2013
3. Jan 24, 2013

### manuel huant

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)) <$\infty$

4. Jan 24, 2013

### Dick

Define a set A(f,n)={x: |f(x)|>1/n}. A(f,n) will have finite measure, right?