# Alternative definitions of outer Lebesgue measure

We have defined the outer measure (in $$\Bbb{R}^n$$) in class as:

$$\mu (A) = \inf \left\{ \sum_{i=1}^\infty v(I_i)\quad \{I_i\}_{i=1}^\infty \text{ collection of open cubes with } A \subset \bigcup_{i=1}^\infty I_i \right\}.$$

I have proved, as our exercises suggested, that the definition holds if we use $$I_i$$ as open and bounded cubes and also if $$I_i$$ are compact cubes.

However, since the exercise doesn't suggest to do the same with closed (not necessarily bounded) cubes, I suppose it's false.

Is this true? Would you know any counterexample?

• It does not matter whether you use open/closed/compact cubes/rectangles/balls, etc. – Sangchul Lee Oct 24 '18 at 8:10
• If you try and think of a "new" form of approximation, like closed cubes, you should ask yourself - don't these have open cubes inside of them? Didn't I already show that it works in this case? – Prototank Oct 24 '18 at 13:01