What are the advantages of measure in mathematics? I am learning measure theory, and I have come to know that measure is generalization of length, area, volume etc. So why do we need to generalize these and what is the advantages of measure here?

You can really only make sense of "length" if a set is an interval or a union of intervals.  But not every subset of $\Bbb R$ is a union of intervals.  Some sets, like the Canter set for example, can't be written as a union of intervals.  But you can still make sense of how big it is, by defining the Lebesgue measure and seeing that it has measure zero.
