Lebesgue measure sigma algebra Lebesgue measure on sigma algebra, help ...........
Which of the following are sigma algebras? reply with justification please.


*

*All subsets in rational numbers

*{ {0},{1},{0,1} }in space {0,1}

*all intervals [x,y) x,y elements of [0,1] and all their unions in the space [0,1)

*all subsets of [0,1]

*all open subsets in real line(with usual metric)

*all finite subsets and all subsets with finite complement in rationals.


please help thank you.
 A: *

*Not a sigma algebra because if we take any subset from the set of subsets of Q denoted by A, we find that the complement of the set we have taken is irrational which is not present in the set A.  Hence not a sigma algebra. 

*The null set is missing in the sets given. Note that for set theory, the null set is a subset of any set by convention whereas in case of sigma algebras, the null set is a requirement.  Therefore for a set to be a sigma algebra, the null set must be there.  It is not a convention but a requirement for a set to be a sigma algebra.
A: *

*

*Revision: Null set is missing, so this is not a $\sigma$-algebra.  Thanks, Arthur.

*The collection you describe generates the Borel Sets.  Is there a Borel set that is not in this collection. 

*Yes.

*Intersect $(-1/n, 1/n)$ and you have your answer.

*this is not a $\sigma$-algebra.  Enumerate the rationals and let $A$ be all elements with even index and $B$ be all elements with odd index.  This lies in the $\sigma$-algebra generated by the finite and cofinite subsets of the rationals, but it is neither finite nor cofinite.
