# Thread: Borel Fields

1. ## Borel Fields

Hi, I was wondering if someone could help me with this. Say we let $\displaystyle {B}({R})=\sigma(\{(a,b]:a<b\})$ be the Borel field of $\displaystyle {R}$.

If we let $\displaystyle {P},{Q}:{B}({R})\rightarrow[0,1]$ be probability measures. How does one prove or disprove that if $\displaystyle {P}((a,\infty))={Q}((a,\infty))$ for all $\displaystyle {a}\in{R}$ then $\displaystyle {P}({B})={Q}({B})$ for all $\displaystyle {B}\in{B}({R})$?

Any advice will be great.

2. Hello,

Do you know Dynkin's theorem that states that if two sigma-finite measures are equal for any element of a $\displaystyle \pi$-system that generates a sigma-algebra, then they're equal for any element of that sigma-algebra ?