I take taking an introductory course on probability and stochastic processes and want to understand the sigma field on R. What is the largest sigma-field on R ? And also what is the Borel sigma field?
Omega (Sample Space) = R F' = {phi, omega, (a,inf), (a,b], (-inf,a), [a,inf), (a,b), [a,b], [a,b), {a} } Is F' a borel sigma field?
Please help.
What do you mean by largest? One that contains most sets? Well, you can take the power set of $\mathbb{R}$, it is a sigma field. (though a not so interesting one, it contains way too many sets, so it's difficult to define an interesting measure on it)
As for Borel sigma field, this is the smallest sigma field which contains all the open subsets of $\mathbb{R}$. A formal definition-the intersection of all sigma fields on $\mathbb{R}$ which contain all the open sets. (it is easy to check that any intersection of sigma fields is also a sigma field). Now this is really an important sigma field-it isn't too large but still contains all the "good sets".