TITLE: What is set of measure zero?
QUESTION [7 upvotes]: What is the concept of set of measure zero? Please explain it in a easy language. Thank you.

REPLY [7 votes]: From an application point of view, a set of measure zero has the property that you can change the value of the function at points in the set without affecting the value of the integral of the function.

REPLY [2 votes]: A set of measure zero  $E$ (under the Lebesgue measure in $\Bbb R^n$ has the following property, for any small $\epsilon$ I choose, there exists countably many cubes $I_k$ (intervals in one dimension) with $E\subseteq I_k$ and $\sum v(I_k)< \epsilon$ where $v(I_k)$ is just the volume of $I_k$.
To truly have any understanding of a measure zero set, you need to understand this definition and the crucial examples that come immediately afterwards. First, note any countable (and hence finite) subset of $\Bbb R^n$ is measure zero. The Cantor set shows there are uncountable measure zero sets. In higher dimensions, for any set $E\subseteq \Bbb R^{n-k}$ with measure $0$  , we have  $E\times \Bbb R^k$  has measure zero in $\Bbb R^n$.