Hello, I have 2 questions:

1. Let Fi be an increasing sequence of sigma fields and define F=UFi ( union from i=0 to infinity of Fi).

I proved that F is a field, and I see that it isn't a sigma field, but I can't find a counter example.

2. Let (Omega, F, P) be a probability space, where

Omega= {(x, y) in R^2 : x^2 + y^2 <= 1},

F the class of all Borel subsets of Omega, and P the probability measure on Omega defined according to the Lebesgue measure.

(a) What is the probability of the event

Ar = {(x, y) : sqrt( x^2 + y^2) <= r}?

(b) What is the probability that the angle between the line connecting a point and the origin, and the positive part of the x-axis, be smaller than alpha, alpha in (−pi, pi]?