A few questions, hopefully to spark some discussion.


How can one define a product of measures?


1. We could use Colombeau products by embedding the measures into the distributions? I'm not sure why this approach is frowned on?
2. 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.
3. 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.


Thanks,


Daniel



Opinions on the Multiplication of Measures
Counterexample for suggestion 2: Take a measure on a 2-point set, assigning to each singleton measure 1 and therefore assigning to the whole set measure 2. The pointwise product of this measure with itself still gives the singletons measure 1 but gives the whole space measure 4, so it's not a measure.


For suggestion 3: There are measurable bijections from $R^2$ to $R$ when $R$ is the real line, but under the "natural" such bijections (e.g., interleaving binary expansions) the image of the product measure (Lebesgue measure times itself, on the plane) would just be Lebesgue measure on the line. By using other, specially constructed bijections, you could get the image to be any of lots of other measures on the line.


