A few questions, hopefully to spark some discussion.
How can one define a product of measures?
- We could use Colombeau products by embedding the measures into the distributions? I'm not sure why this approach is frowned on?
- Since measures are map from sets into numbers, can we not define the product as the pointwise product on sets? I would like to see the counter example here.
- Another product I can think of is to take the product measure and then use the fact that there exists a bijection from R to R^2. I guess that there exist no measurable bijection, which is why that might not work.