let A be the collection of all bounded open intervals on R, B be the collection of all open subsets of R and C be the collection of all closed subsets of R. Then show that the sigma algebras generated by A, B and C are equal to that of each other
To show that the sigma algebras generated by A, B, and C are equal, we need to show that each sigma algebra contains the other two sigma algebras, and thus they are all equal.
First, let's consider the sigma algebra generated by A, denoted by σ(A). Since A contains all bounded open intervals on R, we can easily see that it also contains all open subsets of R, and hence B ⊆ σ(A). Similarly, since every closed subset of R can be written as the complement of an open subset of R, C also ⊆ σ(A).
Next, let's consider the sigma algebra generated by B, denoted by σ(B). Since B contains all open subsets of R, it also contains all bounded open intervals on R, and hence A ⊆ σ(B). In addition, every closed subset of R can be written as the complement of an open subset of R, and thus C ⊆ σ(B) as well.
Finally, let's consider the sigma algebra generated by C, denoted by σ(C). Since C contains all closed subsets of R, it also contains all complements of closed subsets of R, which are exactly the open subsets of R. Therefore, B ⊆ σ(C). Moreover, we can write every bounded open interval on R as a union of closed intervals, and each closed interval is a closed subset of R, so A ⊆ σ(C).
Thus, we have shown that each of the sigma algebras generated by A, B, and C contain the other two sigma algebras, and hence they are equal to each other.
To show that the sigma algebras generated by A, B, and C are equal to each other, we need to prove two things:
1. The sigma algebra generated by A is equal to the sigma algebra generated by B.
2. The sigma algebra generated by A is equal to the sigma algebra generated by C.
Let's prove these two statements one by one:
1. Sigma algebra generated by A is equal to the sigma algebra generated by B:
To show this, we need to prove two inclusions:
a. Sigma algebra generated by A is a subset of the sigma algebra generated by B:
Since every bounded open interval is an open subset, it follows that A is a subset of B. Therefore, the sigma algebra generated by A is also a subset of the sigma algebra generated by B.
b. Sigma algebra generated by B is a subset of the sigma algebra generated by A:
To prove this, we need to show that any open subset of R can be written as a union of bounded open intervals. Let U be an open subset of R. For each x in U, we can find an open interval (a, b) such that x is in (a, b) and (a, b) is contained in U. Then, U can be expressed as the union of all such intervals.
Therefore, the sigma algebra generated by B is a subset of the sigma algebra generated by A.
Since both inclusions hold, we conclude that the sigma algebra generated by A is equal to the sigma algebra generated by B.
2. Sigma algebra generated by A is equal to the sigma algebra generated by C:
To show this, we need to prove two inclusions:
a. Sigma algebra generated by A is a subset of the sigma algebra generated by C:
Since every bounded open interval is not a closed set, it follows that A is not a subset of C. However, we can show that every bounded open interval can be written as the complement of a closed set. For example, (a, b) can be written as the complement of the closed interval [a, b] union [b, ∞) union (-∞, a). Since sigma algebras are closed under complements, it follows that the sigma algebra generated by A is indeed a subset of the sigma algebra generated by C.
b. Sigma algebra generated by C is a subset of the sigma algebra generated by A:
To prove this, we need to show that any closed subset of R can be written as the complement of a union of bounded open intervals. Let F be a closed subset of R. For each x in the complement of F, we can find a bounded open interval (a, b) such that x is in (a, b) and (a, b) is contained in the complement of F. Then, the complement of F can be expressed as the union of all such intervals. Since sigma algebras are closed under complements, it follows that the sigma algebra generated by C is indeed a subset of the sigma algebra generated by A.
Since both inclusions hold, we conclude that the sigma algebra generated by A is equal to the sigma algebra generated by C.
Therefore, we have shown that the sigma algebras generated by A, B, and C are equal to each other.