Convergence of a Sequence: Point-wise, in Measure, Lp, and Uniformly?

[Elenamath]

consider the sequence
fn(x)=n when 0<=x<=1/n
=0 else

Does fn converges:
1. point-wise a.e.
2. in measure
3. in Lp
4. weakly in Lp
5. uniformly

 

[mathman]

1. yes, 3 & 5, no. I have forgotten the definitions for 2 and 4.

 

[Elenamath]

How did you get that 3 and 5 are not true?

For 2: http://en.wikipedia.org/wiki/Convergence_in_measure

For 4: a sequence fn in Lp converges weakly in Lp to f in Lp if:
\int f_n g converges to \int fg, for any g in Lp', 1/p+1/p'=1.

 

[mathman]

The limit function f is 0 (the delta function is not a legitimate function).
For 3, you have ||fn||=||fn-f||=1. Therefore no convergence.
For 5, the problem is what happens at 0 cannot be made uniform.

For 4, it looks like you can have a convergence if you widen the class to include distributions, so that fn -> delta function at 0.

For 2, it looks like it is true. The interval around 0 where fn differs from f can be made as small as you want.

 

[g_edgar]

It says Lp not L1 so the answer may change depending on p, right?

 

[mathman]

No, my assertions are for all p, although ||fn||=||fn-f||=n^(p-1)/p for Lp.