Generating Algebras of Sets
We might not always want to lay out an entire algebra of sets in one go. Sometimes we can get away with a smaller collection that tells us everything we need to know.
Suppose that 
is a subset of 
— a collection of subsets of 
— and define 
to be the collection of finite disjoint unions of subsets in . If we impose the following three conditions on :
- The empty set
and the whole space are both in .
- If
and 
are in , then so is their intersection 
.
- If and are in , then their difference
is in 
then is an algebra of sets.
If 
, then 
, and so contains and . We can also find 
, since 
.
Let’s take 
and 
to be two sets in , written as finite disjoint unions of sets in . Then their intersection is
Each of the 
is in , as an intersection of two sets in , and no two of them can intersect. Thus finite intersections of sets in are again in .
If 
, then 
. Since each of the 
are in , their (finite) intersection 
must be as well, and is closed under complements.
And so we can find that if 
and 
are in , then 
and 
are both in , and is thus an algebra of sets.
March 16, 2010 -
Posted by John Armstrong |
Analysis, Measure Theory
The Unapologetic Mathematician 
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