# Counting measure

1. Dec 1, 2008

### onthetopo

1. The problem statement, all variables and given/known data
In the measure space {X,S,u} where u is the counting measure
X=(1,2,3,..}
S= all subsets of X
fn(x)=$$\chi$${1,2,,,..n}(x) where $$\chi$$ is the characteristic (indicator) function.

Does fn(x) converge
a.pointwise
b.almost uniformly
c.in measure

2. Relevant equations

3. The attempt at a solution
My guess would be
a.pointwise.yes, since it goes eventually to 1 , but it's hard to demonstrate this
b. almost uniformly : yes?
c. in measure: yes? follows directly from b if the answer to b is yes

2. Dec 1, 2008

### rochfor1

a) Yes. Take $$x \in X$$. It shouldn't too difficult to show that eventually $$|f_n ( x ) - 1 | = 0$$.
b) See (c)
c) No. It's easy to see that the only possible limit is $$f\equiv1$$. Now, for any $$1>\varepsilon>0, n \in \mathbb{N}$$, $$\{ | f_n - 1 | > \varepsilon \} = \{ n + 1, n + 2, \ldots \}$$. What is the measure of this set? What does this say about almost uniform convergence?

3. Dec 1, 2008

### onthetopo

I think the only problem is that we have to find the cardinality of (n+1,n+2,n+3...) as n goes to infinity. As n goes to infinity, there is no number larger than n and in fact no n+1,n+2.....can exist?

4. Dec 1, 2008

### rochfor1

That's not a very precise way to think about it. Think about it...without a doubt $$n\to\infty$$, but at any "stage" of this limit, $$n<\infty$$ so the set I wrote about is well-defined, and can in fact be mapped bijectively to the natural numbers. What does that imply about its cardnality?

5. Dec 1, 2008

### onthetopo

I completely understand the solution now.

One last question
How to prove that the metric space (L,d)
where L=all measurable functions
d(f,g)=$$\int{\frac{|f-g|}{1+|f-g|}$$ is complete?

I really have no idea how to show this since I have to show that the limit of ARBITRARY cauchy sequence is another measurable function. I think it is very difficult.

Last edited: Dec 1, 2008
6. Dec 1, 2008

### Dick

Is that the complete question? What domain are your measurable functions defined on?

7. Dec 1, 2008

### onthetopo

Sorry, my bad (X,S,u) is a FINITE measure space and L is the set of FINITE measurable functions. no information other than that.
no mention of lebesgue measure or borel set

8. Dec 1, 2008

### Dick

That helps. Otherwise you wouldn't have a metric. Now your metric is equivalent to the L_1 metric, isn't it? Sorry, it's been a long time since my Real Analysis classes and I don't have this stuff at the tip of my tongue anymore.