In article <ztWdna8Ga43eGvDVnZ2dnUVZ_gGdnZ2d@[EMAIL PROTECTED]
>,
sto <sto@[EMAIL PROTECTED]
> 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_.
>
>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).
>
>
>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.
>
>
>How do you prove that the class _C_ is a sigma-ring? (this is supposed
>to be "easy")
You prove that it is closed under differences and under countable
unions, of course.
> I managed to prove that the simpler class {diff(E,A):E in
>_S_(_E_)} is a sigma-ring, but can't find any way to prove that _C_
>itself is a sigma-ring.
An arbitrary element of _C_ is, as you note, the union of B_1 in
_S_(int(_E_,A)) and (E_1-A) for some E_1 in _S_(_E_).
So to show _C_ is closed under differences, you consider
(B_1 \/ (E_1-A)) - (B_2 \/ (E_2-A)
for some B_1, B_2 in _S_(int(_E_,A)), and some E_1,E_2 in
_S_(_E_). Try to express it as the union of something in
_S_(in(_E_,A)) and some (E'-A) for E' in _S_(_E_). You'll want to use
the fact that the elements are in specific sigma rings.
The closure under countable unions is simpler, since if you have a
family {B_i \/ (E_i-A)} i=1,2,3,... then the union of the family is
just (\/ B_i) \/ (\/E_i - A), and now you can use the fact that the
B_i live in a sigma ring and the E_i live in a sigma ring to deduce
this is of the desired form.
--
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"It's not denial. I'm just very selective about
what I accept as reality."
--- Calvin ("Calvin and Hobbes" by Bill Watterson)
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Arturo Magidin
magidin-at-member-ams-org
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