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!?