1. To write out G = o(A1, A2, A3), we need to find the smallest σ-algebra containing the sets A1, A2, and A3.
Since A1 = {1, 2, 3}, A2 = {3, 4}, and A3 = {4, 5, ..., n}, we can see that A2 is a subset of A1, and A3 is a superset of A2.
Therefore, G = o(A1, A2, A3) will include all subsets of A1, A2, and A3, as well as any unions and intersections of these sets.
Now, let's consider whether f(x) is measurable with respect to G. For a function to be measurable with respect to a σ-algebra, the pre-image of any measurable set must be measurable.
In this case, since f(x) is defined as 0 for x = 1, 2, ..., n-1, and 1 for x = n, the pre-image of any set containing 0 will be a measurable set.
Therefore, f(x) is measurable with respect to G.
Examples of o-algebras that f is measurable with respect to could be:
1. The power set of {1, 2, ..., n} (the set of all subsets of {1, 2, ..., n}).
2. The empty set and the set {n}.
(b)
2. To show that every partition on Ω corresponds to an o-algebra on Ω, we need to demonstrate that the collection of sets in the partition satisfies the properties of an o-algebra.
A partition on Ω, denoted as S = {S1, S2, ..., Sn}, consists of sets Si that satisfy the following conditions:
1. Si ∩ Sj = Ø for all i ≠ j. (No two sets in the partition overlap.)
2. Ω = U Si, where U represents the union symbol. (The union of all sets in the partition covers the entire sample space.)
To show that this partition corresponds to an o-algebra, we need to prove that the collection of sets in the partition satisfies the following properties:
1. The empty set and the sample space Ω are in the collection.
2. The collection is closed under complements. That is, if A is in the collection, then its complement, Ω - A, is also in the collection.
3. The collection is closed under countable unions. That is, if A1, A2, A3, ... are sets in the collection, then their union, U Ai, is also in the collection.
Let's go through these properties:
1. Since the partition covers the entire sample space, Ω is in the collection. Additionally, since the sets in the partition are disjoint, their intersection will be the empty set. Therefore, the empty set is also in the collection.
2. For any set A in the collection, we know that A is one of the sets in the partition. Taking the complement of A will include the elements that are not in A, which will be covered by the other sets in the partition. Therefore, the complement Ω - A is also in the collection.
3. If we have a countable sequence of sets in the collection, their union will include all the elements covered by each set. Since the sets in the partition cover the entire sample space, their union will be Ω, which is also in the collection.
By satisfying these properties, we have shown that every partition on Ω corresponds to an o-algebra on Ω.
Have an amazing day :)