Re: Null Sets and Product Measures.

On 1 Apr 2007 22:56:44 -0700, "jestingrabbit@xxxxxxxxx"
<jestingrabbit@xxxxxxxxx> wrote:

My question is about the measurability of a class of subsets of a
product space. Consider X=[0,1]^2 with the standard measure ie
measurable sets generated by sets of the form [a,b]*[c,d] and with
measure m such that m([a,b]*[c,d])=(b-a)(d-c).

For every 0<=x<=1, let there be a set A(x) which is a subset of [0,1]
with measure 0. Define B={(x,y) in X | y is in A(x)}.

If we can use Fubini's theorem then we can conclude that m(B)=0.
However, I'm not sure if B is measurable, and if its not then I don't
think I can use Fubini's. The function A isn't measurable in any
obvious sense, and B seems to not be measurable in the uncompleted
product sigma algebra. Is it measurable in the completed product sigma
algebra? Can I use Fubini's?

You can find the counterexample various people have mentioned
in Rudin "Real and Complex Analysis".

It seems quite possible that your set B _is_ measurable,
because of things you haven't told us. A person has to
do some work to prove the existence of a non-measurable
set (for example it's impossible to prove it without
using the Axiom of Choice) - if your sets A(x) arise
from some "explicit" "construction" then it does seem
to me quite likely that B is measurable, even though
that doesn't follow from what's above.


************************

David C. Ullrich

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