Are Sigma Algebras Unique for a Given Set?

  • Thread starter ChemEng1
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  • #1
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Main Question or Discussion Point

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.
 

Answers and Replies

  • #2
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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.
 
  • #3
Bacle2
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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.
 
  • #4
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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."
 

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