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?
 A: 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.
A: Measures provide a natural way to extend integration onto various spaces. (manifolds, reals, integers, etc.) Given a measure space $(X,\Sigma,\mu)$, a simple function is the finite sum of characteristic functions $\phi = \sum_{j=0}^{m-1} \chi_{S_j}$, where $\bigcup_{j=0}^{m-1} S_j = X$ and they are mutually disjoint. $\phi$ can be assigned a integral
$$ \int \phi\,d\mu = \sum_{j=0}^{m-1} \mu(S_j)
$$
Based on this, any nonnegative measurable function $f:X \to [0,\infty]$ can be assigned a integral
$$ \int f\,d\mu = \sup\left\{ \int \phi\,d\mu \mid \phi \leq f, \phi \text{ simple} \right\}, \int_S f\,d\mu = \int f\chi_S\,d\mu
$$
On $\mathbb{R}^n$, the most important measure is the Lebesgue measure $\lambda$. It turns out to coincide with the usual geometric definition of area/volume, and with the Riemann integral
$$ \int_{[a,b]} f\,d\lambda = \int_a^b f\,dx
$$
Measures also provide a way to unify discrete and continuous summation. Consider the counting measure $\mu$, which evaluates to the number of elements in a set. Then for any sequence $\{x_j\}_{j=0}^\infty$,
$$ \sum_{j=0}^\infty x_j = \int x\,d\mu
$$
A: The point of introducing the notion of measure is to generalise the notion of an integral.
The Riemann integral has a relatively straightforward definition, but has poor behaviour in limiting arguments; whereas the Lesbegue integral behaves much better, and the main theorems - the monotone and dominated convergence theorems, as well as Fubini - can be seen as analogues for sequences and series.
In fact, by choosing the right measure these theorems become exactly those theorems.
Further, we can integrate over abstract spaces, such as groups - by the Haar measure - and this helps in generalising, say the representation theory of finite abelian groups to compact abelian groups.
