Elements of a(C) 
*

*Denote by C the system 
{ (a,b] : -$\infty < a \leq b < \infty $} $\bigcup$ { (a,$\infty $) a $\in \mathbb R$ } $\bigcup$ {(-$\infty $,b] :  b $\in \mathbb R$}
Remark that $\mathbb R$ is not in C. Can someone explain why this is true? 

*Let A be an arbitrary elemtn of the algebra a(C) Give a representation of A in terms of elements of C ; 
I wrote 
A = { $\bigcup${i=1...n}$\ A_i$ where $\ A_i \in C $ for all i=1,..n}
Is this right? 


*

*Is this representation unique, yes or no? A. no 


And finally, this is where I'm really stuck;
Let F: $\mathbb R \rightarrow \mathbb R $ be a right continuous and non decreasing function. Define $\ P_0 : C\rightarrow [0,\infty]$  by $\ P_0(I) = F(b)-F(a) : I = $ {$\ (a,b] -\infty < a \leq b < \infty $} , $\ F(\infty) - F(a) : I = (a,\infty) a\in \mathbb R $ , $\ F(b)-F(-\infty) : I = (\infty,b]  b\in \mathbb R $. Where $\ F(\infty) = sup$ {$\ F(x): x \in \mathbb R $} and $\ F(-\infty) = inf$ {$\ F(x): x \in \mathbb R $}
Construct a content on ($\mathbb R $, a(C)) which coincides with $\ P_0 $ on C ?? 
How would you do this when you dont know F!?
 A: If $\mathbb{R} \in C$, then there would be real numbers $a$ and $b$ such that $\mathbb{R} = (a,b]$ or $\mathbb{R} = (a,\infty)$ or $\mathbb{R} = (-\infty, b]$. It's easy to see that each of these is impossible. Just check a few numbers to see why yourself.
Your representation is okay, but notice that the union of any two non-disjoint (i.e. overlapping) elements of $C$ is again in $C$. So the representation is not unique (since you can arbitrarily split up any of the elements of $C$ into the disjoint union of two elements of $C$). So you can do slightly better: you can impose a stronger condition on your representation of $A$. Hint: The condition I'm referring to is alluded to in this very paragraph; also the condition will make defining a content on $A$ easier.
You will indeed have to define your content in terms of $F$. The point is that $P_{0}$ defines a set function only on $C$, so you need to extend $P_0$ to the algebra $A$, meanwhile retaining the properties necessary to yield a content. As my hint above said, the "right" representation of $A$ will make this easier.
