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How to prove this statement?

"Any set of measure $0$ can be covered by a sequence $((a_n,b_n))$ of intervals with $\sum (b_n-a_n) <\epsilon$."

It seems like this statement is correct by intuition. but I do not know how to prove it, could anyone help me in doing so?

Emptymind
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    It's the definition, is it not? –  Dec 08 '19 at 10:15
  • which [email protected]. – Emptymind Dec 08 '19 at 10:16
  • The definition of Lebesgue measure: the Lebesgue outer measure of a set is the greatest lower bound of those sums taken over all the coverings by intervals. If the set also happens to satisfy certain conditions, then we say that the set is measurable and that the aforementioned number is its Lebesgue measure. –  Dec 08 '19 at 10:17
  • And why the outer measure is the same as the measure in our [email protected]. – Emptymind Dec 08 '19 at 10:21
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    Because I said so in my second sentence. –  Dec 08 '19 at 10:22
  • You should give us your definition of "measure 0". – Paul Frost Dec 08 '19 at 10:27
  • I am confused I am using Royden 4th edition , "real analysis" and I do not remember where is the definition of "measure 0", could you please tell me if you know? @PaulFrost – Emptymind Dec 08 '19 at 10:34
  • I don't know the book, thus I can't help. But it seems that the statement in your question is not the definition. – Paul Frost Dec 08 '19 at 10:37
  • @Emptymind Look at the definition of Lebesgue measure (presumably in the chapter that deals with Lebesgue measure, not the one that treats measure theory in general). –  Dec 08 '19 at 14:06

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