Let [tex]\Omega = [0,1)[/tex]
Let G be the collection of all subsets of [tex]\Omega[/tex] of the form
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
Let [tex]\Omega[/tex] be any set. A collection F of subsets of [tex]\Omega[/tex]
is a field if:
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
The Attempt at a Solution
To show G is a field:
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.
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.