Convergence of lebesgue integral

[onthetopo]

Homework Statement

$$fn(x)=n\chi_{(0,n^{-1})}(x)-n\chi_{(-n^{-1},0)}(x)$$ -1<x<1
Prove that $$\int fn(x)dx\rightarrow\int f(x)$$
a. Using direct computation
b. Use dominated convergence

The Attempt at a Solution

a. How to do direct computation?

b. Using dominated convergence
If x is in (0,1/n), then fn(x)=<1-0=1
If x is in (-1/n,0), then fn(x)=<0-1=-1
If x is in neither , then fn(x)=0
Thus |fn(x)| is dominated by |g| where g(x)=1 , a constant function on (-1,1)
g(x) is in L1, since g(x) is Riemann integreable on (-1,1) it must also be Lebesgue integreable.
Thus, by dominated convergence theorem:
"limit of integration of fn" is equal to "integration of limit of fn"= integration of f(x)=0 as n goes to +infinity.
Thus,

$$\int fn(x)dx\rightarrow\int f(x)\rightarrow0$$ QED

 

[morphism]

$\int f_n$ isn't that hard to comptue... What have you tried?

 

[onthetopo]

Can we say that
$$\int fn(x)=\int n\chi_{(0,n^{-1})}(x) +\int -n\chi_{(-n^{-1},0)}(x)$$ -1<x<1 (**)
I think the above is the crucial step. What exactly do we need for the above to hold? I guess as long as each term on the right is integreable?

Assume (**) holds, then let u be the lebesgue measure
$$\int n\chi_{(0,n^{-1})}(x) +\int -n\chi_{(-n^{-1},0)}(x)=n*u(0,n^{-1})-n*u(-n^{-1},0)=n*1/n-n*(1/n)=0$$

Is my answer to part (a) posted here , together with my answer to part (b) posted in the first post,both completely correct? =)

 

[onthetopo]

Can someone please tell me if what i did is right?

 

[morphism]

Yup, that's good.

Your answer for part (b) is incorrect however. f_n isn't dominated by 1. Try to sketch f_1, f_2, and f_3 to get a feel of what's going on.

 

[onthetopo]

If x is in (0,1/n), then fn(x)<n*1/n-0=1
If x is in (-1/n,0), then fn(x)>0-n*1/n=-1
If x is in neither , then fn(x)=0
Thus |fn(x)| is dominated by |g| where g(x)=1, I really can't see what's wrong?

 

[morphism]

No, if x is in (0,1/n), then f_n(x)=n; and if x is in (-1/n,0), then f_n(x)=-n.

 

[onthetopo]

ah!
so it's dominated by |n|, which is not in L1, so we can;t use DCT at all

 

[morphism]

What do you mean?

What you want to do is find a nice integrable function g on [-1,1] that dominates all the f_n's. It doesn't look like you've tried to do this.

 

[onthetopo]

if x is in (0,1/n), then f_n(x)=n; and if x is in (-1/n,0), then f_n(x)=-n., and if x is in neither, f_n(x)=0
thus, |f_n(x)|<|n| for all x on (-1,1) , but|n| is unbounded as n goes to infinity
we cannot find a g(x) in L1 such that |fn|<|g| for all n and all x in (-1,1).

 

[morphism]

That doesn't mean you can't fit a g in between f_n and n. I don't immediately see a good candidate though; and I'm willing to guess that you're right, dct probably won't work. You can use a generalized version of the dct (say, the one mentioned in your other thread), but that's kind of stupid though, because it amounts to basically doing what you did for part (a).