# Show that X ⊂ ℜn has measure 0 if and only if ε > 0

1. Dec 3, 2011

### Riam

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

Please I need your help in this question. I don't know how to answer it.

The question: Show that X ⊂ ℜn has measure 0 if and only if ε > 0 there exists an infinite sequence of balls

B_i ={ x ∈ R^n| |x-a_i | < r_i} with ∑ r$^{n}_{i}$ < ε such that X ⊂ ∪ $^{\infty}_{i =1}$B_i

2. Relevant equations

3. The attempt at a solution

2. Dec 3, 2011

### gb7nash

Re: measures

What have you tried?

3. Dec 3, 2011

### micromass

Staff Emeritus
Re: measures

Please post an attempt at the solution or this thread will be deleted.

Also it might be necessary to define your terms. How did you define "measure 0" etc.

4. Dec 3, 2011

### Riam

Re: measures

I said
choose ε > 0, , for n = 1, i = 1, let a be the center of the ball and raduis r. if | x- a| < r with Ʃ r < ε such that X $\subset$ B$_{1}$. and keep trying for n =2 and generalise it? this is my guess?

5. Dec 3, 2011

### micromass

Staff Emeritus
Re: measures

And how did you define "measure 0"??