Opinions on the Multiplication of Measures

Question

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

Answer 1

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.

Answer 2

Not a definitive answer, just some thoughts that I hope you might find useful. Firstly, there is a celebrated example of Schwartz that shows that you can't do this globally in the sense of getting a ring structure on the space of measures which extends that on the space of continuous functions. It can, however, often be fruitful to try a local approach, i.e., find pairs of measures which {\it can} be multiplied in a sensible way. For example, the product of two dirac measures is only a problem if their singularities coincide. One can, trivially, always multiply a continuous function and a measure (and with a little bit of integration theory this can be extended to more general functions). One way to deal with less trivial situations is to say that if we can cover $I$ (for the sake of simplicity, I will assume that the measures are defined on the unit interval $I$) with a finite family of relatively open subintervals so that on each interval at least one of the the measures is a continuous function (which one depending on the interval, of course), then we can define the product. This covers the above case of the product of dirac measures with distinct singularities. One would have to know more about the kind of measures that you want to multiply in order to decide if this is of much help.