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") 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.
sto <s...@address.invalid> 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.
-- ====================================================================== "It's not denial. I'm just very selective about what I accept as reality." --- Calvin ("Calvin and Hobbes" by Bill Watterson) ======================================================================
Arturo Magidin wrote: > In article <ztWdna8Ga43eGvDVnZ2dnUVZ_gGdn...@earthlink.com>, > sto <s...@address.invalid> 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.
This is exactly the approach I took originally, but I keep running into the problem that
(B1 \/ E1 - A) - (B2 \/ E2 - A)
reduces to
(B1 - B2) - (E2 - A) \/ [(E1 - E2) - A] - B2
Of course the B1 - B2, E1 - E2, and even [(E1 - E2) - A] terms belong to their respective sigma-fields, but in the end I don't see that the expression reduces to the form B \/ E - A for some B in _S_(int(_E_,A)) and E in _S_(_E_). I've been checking my algebra all day. Maybe I've just been without sleep too long, but I can't see how to prove it this way.
I wonder whether there isn't some deeper significance to the fact that
E = E /\ A \/ E - A
and the fact that each element of _C_ is the union of one element from _S_(int(_E_,A)) and one from _S_(_E_-A)?
> 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.
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)
In article <gbadnbfpfIh2APDVnZ2dnUVZ_rrin...@earthlink.com>,
sto <s...@address.invalid> wrote: >Arturo Magidin wrote: >> In article <ztWdna8Ga43eGvDVnZ2dnUVZ_gGdn...@earthlink.com>, >> sto <s...@address.invalid> wrote:
>>> 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.
>This is exactly the approach I took originally, but I keep running into >the problem that
>(B1 \/ E1 - A) - (B2 \/ E2 - A)
>reduces to
>(B1 - B2) - (E2 - A) \/ [(E1 - E2) - A] - B2
You'll obviously want to rewrite it in some way. Just doing the opeartions will not be good enough. You can try decomposing some of these terms further, naturally.
>Of course the B1 - B2, E1 - E2, and even [(E1 - E2) - A] terms belong to >their respective sigma-fields, but in the end I don't see that the >expression reduces to the form B \/ E - A for some B in _S_(int(_E_,A)) >and E in _S_(_E_). I've been checking my algebra all day. Maybe I've >just been without sleep too long, but I can't see how to prove it this way.
There seems to be another reply that shows how to decompose this.
>I wonder whether there isn't some deeper significance to the fact that
>E = E /\ A \/ E - A
>and the fact that each element of _C_ is the union of one element from >_S_(int(_E_,A)) and one from _S_(_E_-A)?
Quite possibly...
-- ====================================================================== "It's not denial. I'm just very selective about what I accept as reality." --- Calvin ("Calvin and Hobbes" by Bill Watterson) ======================================================================
William Elliot wrote: > 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)
