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  1. 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?

  2. 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?

  1. 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!?

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    @Rosie: A fairly precise citation (even if not on the internet) would help to allay the concerns of those who wish to avoid giving answers to homework problems.2012-05-21

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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.

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    @Rosie Yes, the condition that the $A_i$'s be pairwise disjoint is exactly what I was referring to.2012-05-21