In the context of Lebesgue measure on $\mathbb{R}^n$, is null set the same thing as a set of measure zero?
I understand that null set implies measure zero, not sure about the other direction.
Update: By null set I mean https://en.m.wikipedia.org/wiki/Null_set a set that can be covered by a countable union of intervals(balls) of arbitrarily small total length.
To be precise: Is a set with measure zero coverable by countable union of balls of arbitrarily small total length?
Best Answer
Yes. This follows from the fact that the Lebesgue measure is constructed using a generator of the Borel $\sigma$-algebra (say the set of open intervals in $\mathbb{R}$) using Caratheodory's Extension Theorem.
In particular, if we denote $\mu((a,b)) = b-a$ for any open interval $(a,b)$, we can define for any $A\subset\mathbb{R}$
$$\lambda^*(A) =\inf\left\{\sum_{n=1}^\infty\mu(A_n): \text{each $A_n$ an open interval, $A \subset \bigcup_{n=1}^\infty A_n, $}\right\}.$$
This $\lambda^*$ is the Lebesgue outer measure. If $\lambda$ denotes the Lebesgue measure then any measurable set satisfies $\lambda(A) = \lambda^*(A)$. The result then follows by the definition of the infimum.