I know that by definition of Lebesgue measure, if the set N has Lebesgue measure zero i.e. m(N) = 0 then we can cover the set by countably many open intervals {a$_{j}$, b$_{j}$}$_{j\in\mathbb{N}}$ such that $\sum_{j}$ b$_{j}$ - a$_{j}$ $<$ $\epsilon$.

I was wondering if the set N is contained in some bounded interval [a,b], would I be able to replace the countable open cover by finitely many open (left open or right open) intervals.

In other words, is it possible to say that I can cover N by finitely many intervals {a$_{j}$, b$_{j}$}$_{j=1}^{n}$ such that $\sum_{j=1}^{n}$ b$_{j}$ - a$_{j}$ $<$ $\epsilon$.


The quick answer is no.

Here is an interesting exercise: see if you can prove that if $N$ is the set of rational numbers in between $0$ and $1$ and $N \subset \bigcup_{k=1}^n (a_k,b_k)$, then $\sum_{k=1}^n (b_k - a_k) \ge 1$.

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