1. The problem statement, all variables and given/known data Let [tex]\Omega = [0,1)[/tex] Let G be the collection of all subsets of [tex]\Omega[/tex] of the form [a1,b1),[tex]\cup[/tex][a2,b2),[tex]\cup[/tex]...[tex]\cup[/tex][ar,br) For r any non-negative integer and 0<=a1 and a1 <=b1 <= a2 .... Show that G is a field Show that G is not a [tex]\sigma[/tex]-field 2. Relevant equations Definition: Let [tex]\Omega[/tex] be any set. A collection F of subsets of [tex]\Omega[/tex] is a field if: 1. [tex]\emptyset[/tex][tex]\in[/tex]F 2. given A in F, then Ac=[tex]\Omega[/tex]\A 3. given A and B in F, A[tex]\cup[/tex]B is in F In addition, the collection F is a [tex]\sigma[/tex]-field if given A1, A2, A3.... are all in F, so is there union [tex]\bigcup[/tex]Ai 3. The attempt at a solution To show G is a field: 1. Assume empty set is not in G, then there must be an element in empty which is not in the G - a contradiction since there are no elements in the empty set. 2. If we have A in G then A=[ai,bi) So Ac= [tex]\Omega[/tex]/A 3. This is where I have the problem, Set A in G as [ai,bi) And B in G as [aj,bj) So we want to show A U B in G But if bi= aj Then we will have A U B =[ai,bj) Which is not in G To show G is not a sigma field: (0,1)= the union of intervals [1/n,1) [1/n,1) is in G for all n (0,1) is not in G So G is not a sigma field. So essentially the problem is with part 3 of the definition.