On Thu, 3 Jul 2008, sto wrote:
> Let X be a set, and define a (Boolean) sigma-ring _S_ as a non-empty
> class of subsets of X that is closed under the formation of differences
> and countable unions.
>
> Let _E_ be any class of subsets of X, and denote by _S_(_E_) the
> smallest sigma-ring containing _E_.
>
Pardon me while I make your notation manageable.
S_E = sigma ring generated by E
> Let A be a subset of X, and denote by E the generic element of the class
> _E_. Denote by intersection(_E_,A) the class of sets
> {intersection(E,A): E in _E_}. Denote by _S_(intersection(_E_,A)) the
> smallest sigma-ring containing the class of sets intersection(_E_,A).
>
E*A = { U /\ A | U in E }
> Based on these definitions, create a class of sets _C_ to be {union(B,
> diff(E,A)): B in _S_(intersection(_E_,A)), E in _S_(_E_)}. In other
> words, each element of the class _C_ is the union of an element B of
> _S_(intersection(_E_,A)) with the difference of an element E of _S_(_E_)
> and the set A.
>
C = { B \/ U\A | B in S_E*A, U in S_E }
Let B \/ U\A and D \/ V\A be two elements of C
(B \/ U\A) - (D \/ V\A) = (B \/ U\A) /\ (X\D /\ (X - V\A))
.. . = (B \/ U\A) /\ X\D /\ (X\V \/ A)
.. . = (B \/ U\A) /\ ((X - D\/V) \/ A\D)
.. . = X\D /\ ((B\V \/ U\(A \/ V) \/ (B /\ A) \/ (U /\ A))
.. . = B\(V \/ D) \/ U\(A \/ D \/ V) \/ ((B /\ A)\D) \/ ((U /\ A)\D)
.. . = B\(V \/ D) \/ U\(A \/ V) \/ (B\D \/ ((U /\ A)\D)
.. . = (B\D - V) \/ (U\V - A) \/ K
Where K = (B\D \/ ((U /\ A)\D) in S_E*A
.. . B\D - V in S_E*A
which upon putting it together, shows C closed under set difference.
Since S_E*A and S_E are sigma rings, it's easy to see
that C includes countable unions of elements of C.
> How do you prove that the class _C_ is a sigma-ring? (this is supposed
> to be "easy")
It isn't when coming to showing closure under set difference.
That's a set algebra grind.
----
|
