Product sigma algebras
According to Folland’s book, the product sigma algebra 
of the product space 
of measurable spaces 
is the smallest sigma algebra for which all the coordinate projections 
are measurable. That is, it is the sigma algebra generated by the sets 
According to PlanetMath’s entry, it is the sigma algebra generated by the sets 
where 
for all but finitely many 
It’s easy to see these definitions coincide when 
is countable– in this case, is the sigma algebra generated by all products of the form 
where 
, but what if is uncountable? It seems reasonable to expect that they would coincide, specifically, that is the sigma algebra generated by all products of the form where and for all but countably many
Any ideas on a proof? Duh, never mind. Don’t you hate it when you ask a question you think is nontrivial if not actually hard, then you realize that you spoke too soon?
Possibly relevant posts:
- Systems of Sets (10/7/2005)
- I’m stumped– you give it a try (4/1/2008)
- Inner products (6/7/2006)