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I'm taking multivariable calculus course (third one actually) and we're introduced to measure zero.

What does one need measure zero for?

Measure zero of a set $A$ in $\mathbb{R}^n$ means that one can find a collection of sets whose volume can be made arbitrarily small and so that $A$ is contained in the sum over such collection.

Why is this an important property?

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    Among other things, because a bounded (on some interval) real function is Riemann-integrable there iff its set of discontinuity points at that interval has Lebesgue measure zero... – 2017-01-22
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    It would help if you gave us a bit more context here. How did you come across the idea of "measure-zero sets" in particular? – 2017-01-22
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    I'd say that the idea of a "measure zero set" is just a piece of measure theory, which doesn't hold much significance outside that context. So, I think a more tractable would be "why do we care about measure theory?" Even for that question, some context would be very helpful. – 2017-01-22
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    [This](http://math.stackexchange.com/q/1286025/212120) is a related question, and the answer is nice but a bit misleading concerning when it talks about measure zero sets as being made of isolated points. So for some more intuition you should peruse [this other question](http://math.stackexchange.com/q/459849/212120). I think that you can browse this site and Wikipedia for some basic information and then edit your question to something more specific. – 2017-01-23

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