TITLE: When is the Lebesgue measure of a set zero?
QUESTION [0 upvotes]: Let $l^n$ be the Lebesgue measure on $ \sigma$-algebra $L^n$ of Lebesgue measurable subsets of $\mathbb{R}^n$ and $A\in L^n$.
Suppose there exists an open covering $\{A_i\}_{i\in\mathbb{N}}$ with finite $\sum_{i=1}^\infty l^n(A_i)$ and $A\subset A_i$ for infinitely many $i$.

How do I show that $l^n(A)=0$?

I really appreciate any hints as where to start, since I don't know how to begin.
I do know that
$$l^n(X)=\sup\{l^n(C):C\subset X,C\text{ compact}\}=\inf\{l^n(O):X\subset O,O\text{ open}\}$$
and I think this must be used.

REPLY [0 votes]: Hint: recall that if a series of positive numbers is finite, then the sequence converges to zero.
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