understanding the properties of a measure space A measure space is a triple $(X,\sum , \mu)$
where 
(i) X is a set.
(ii) $\sum$ is a $\sigma$ algebra of subsets of X.
(iii)$\mu : \sum \to [0,\infty]$ is a function such that 
(a) $\mu \phi =0$
(b) if $<E_n>_{n \in \mathbb{N}}$ is a disjoint sequence in $\sum$, then 
$\mu(\cup _{n\in\mathbb{N}} E_n)=\sum^\infty_{n=1} \mu E_n$
Intuitively, how can $\mu : \sum \to [0,\infty]$ be interpreted? Is it just saying that the function $\mu$ assigns a length between 0 and $\infty$ to a collection of subsets of X?
How can (iii) (b) be interpreted intuitively?  What is it saying and why is it required? How can it be deduced? 
 A: For once the name has been appropriately chosen. It's a measure so it can represent the length of line segments, the area of planar figures, the volume of solids, etc.
Condition (iii)(b) is called countable additivity. It corresponds to the requirement that we should be able to compute the length of an infinite chain of intervals by either subtracting the end points of the chain or by computing the sum of a series and still obtain the same result.
It cannot be deduced from anything, it's a hypothesis.
There is a parallel theory where countable additivity is replaced with the weaker condition of finite additivity (you can guess how to formulate that condition).
The main reason why people prefer the stronger hypothesis is integration theory.
Integrals have better properties in a countably additive theory than in a finitely additive theory.
A: Traditionally, the measure "measures" a size of given set. $\Sigma$, which may be called "$\sigma$-algebra", means the collection of sets which are measurable. 
(iii) (b) is called $\sigma$-additivity. You can adapt the finite-additivity of $\mu$, which states that $\mu(A\cup B) = \mu(A)+\mu(B)$ if $A$ and $B$ are disjoint. $\sigma$-additivity just extend the finite-additivity to infinite sums, but it is quite useful when handling the limit occurring in measure theory.
