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

[Riam]

Homework Statement

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

Homework Equations

The Attempt at a Solution

 

[gb7nash]

What have you tried?

 

[micromass]

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.

 

[Riam]

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?

 

[micromass]

And how did you define "measure 0"??