Is Every Element of a Sigma-Algebra Generated by a Countable Subcollection?

[fourier jr]

Here's what it says:

"Let C be a collection of sets & E an element in the sigma-algebra generated by C. Then there is a countable subcollection C_0 contained in C such that E is an element of the sigma-algebra A_0 generated by C_0."

 

[mathwonk]

i suggest recalling the definition of a sigma algebra.

i.e. for analogy, if A is a polynomial ring generated by a set S, and E is an element of A, then there is a finite subset T of S such that E belongs to the polynomial algebra generated by T.

 

[fourier jr]

here's my definition:
i) empty set is in C
ii) if a set is in C then so is its complement
iii) if a collection of sets is in C then so is the union of all those sets

i guess with de morgan's laws intersections of sets are also in there, but I'm not sure how that helps.

 

[mathwonk]

well it has been over 4 decades for me, but i do not recall your dfiniton as being corrct for sigma algebras. I think there is a countability assumption in there on those unions.

you might check it. yeah, here is what I googled up:

http://mathworld.wolfram.com/Sigma-Algebra.html