Question
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
Answers
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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.
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.