Lebesgue Outer Measure in R: interval in the cover are limited?

Question

We all know the definition of Lebesgue outer measure in $R$.

$ m(E) = inf\{\sum_{k=0}^\infty l(I_k) : E\subset \bigcup_{k=0}^{\infty} I_k\} $

In all the definition I saw, $I_k$ are open intervals.

This means they're limited or they can be also unlimited?

And, another question, is the empty set considered as interval in the definition?

Thanks!

Answer 1

The definition also works with $I_n$ all closed, all half-open, or a mixture of both.

The length function $\ell$ is usually just defined on bounded intervals.

If you want, you can allow unbounded intervals, defining the "length" of an unbounded interval to be $\infty$, but that would be redundant. You want the infimum over all such sums; if one interval is unbounded, then the sum is automatically infinite. $\infty$ is the last "number" you'd want to look at if you are looking for an infimum.

Try to prove that the two quantities:

$$\inf\left\{\sum_n \ell (I_n): \text{the $I_n$'s are intervals covering $A$}\right\}$$

and

$$\inf\left\{\sum_n \ell (I_n): \text{the $I_n$'s are bounded intervals covering $A$}\right\}$$

are equal.