Are Sigma Algebras Unique for a Given Set?

[ChemEng1]

Is there only 1 σ-algebra generated for a set?

Consider M={1,2}. Ʃ(M)={∅,M} satisfies the definition of a σ-algebra. However, Ʃ(M)={∅,{1},{2},{1,2}} also satisfies the definition of a σ-algebra. However, the way that my text presents these problems (Prove that the σ-algebra generated...) implies that they are unique.

Any help would be appreciated.

 

[csopi]

For every set M, there are at least two sigma-algebras over M, namely {∅, M} and P(M), the power set of M, these are called trivial algebras.

However, if N is a subset of P(M), there is a unique smallest sigma-algebra over M, that contains every element of N. This is called the sigma algebra generated by N. I think, this is what your book refers to.

 

[Bacle2]

Is there only 1 σ-algebra generated for a set?

Consider M={1,2}. Ʃ(M)={∅,M} satisfies the definition of a σ-algebra. However, Ʃ(M)={∅,{1},{2},{1,2}} also satisfies the definition of a σ-algebra. However, the way that my text presents these problems (Prove that the σ-algebra generated...) implies that they are unique.

Any help would be appreciated.

Hi, ChemEng1 : like csopi said, given a subset N of M , with N≠∅ and N≠M, there

is only one σ-algebra generated by N . By construction, this will be the intersection

of all σ-algebras containing N. If , instead of N , you had two or more subsets, then the

σ-algebra would contain the union of all sets, their intersection, their complement, etc.

These two definitions of minimality are equivalent. Basically, if N is in the algebra, then

so is its complement. If sets A,B are in the sigma algebra, then so are A\/B , A/\B,

A\B , etc. This extends to 3-or-more sets.

 

[ChemEng1]

Thanks for confirming this. I spoke to the professor about this. His comment was "My native language is Russian. Questions clarifying how I use articles are completely fair."

 

[blue_raver22]

I would say that the uniqueness of sigma algebras for a given set is not always guaranteed. In the example given, both Ʃ(M)={∅,M} and Ʃ(M)={∅,{1},{2},{1,2}} satisfy the definition of a sigma algebra for the set M={1,2}. This is because the definition of a sigma algebra only requires that the set contains the empty set and is closed under countable unions and complements. Therefore, there can be multiple sigma algebras that satisfy these conditions for a given set.

However, it is important to note that the sigma algebra generated for a set is unique in the sense that it is the smallest sigma algebra that contains all the subsets of the given set. In other words, it is the smallest collection of subsets that satisfies the definition of a sigma algebra for the given set. This is why the text presents the problem as proving the uniqueness of the sigma algebra generated for a set.

In summary, while there can be multiple sigma algebras that satisfy the definition for a given set, the sigma algebra generated for a set is unique in the sense that it is the smallest collection of subsets that satisfies the definition of a sigma algebra for that set.