Infinite Product over Real Intervals

Question

Is this a valid mathematical expression? $$\prod_{n\in[a,b]} n$$ I would think that for $n\in[0,b]$, the product equals $0$, and for $n\in[-a,-b]$, it diverges. Is it possible that such a product would converge to an interesting value for something like $n\in[1/\pi,\pi]$, or is the whole idea nonsense? Has anyone developed a rigorous theory concerning this?

Answer 1

Products, like sums, are done over discrete sets to avoid problems with adding uncountably many things (there is no way to make such a theory make sense for real numbers: a sum of uncountably many nonzero reals always diverges.

There is, however, a construction analogous to the integral, but with products: it's unsurprisingly called the product integral; depending on how you define it, it comes out equal to either $$ \exp{\left(\int_a^b \log{f(x)} \, dx \right)} $$ or $$ \exp{\left(\int_a^b f(x) \, dx \right)}; $$ the latter, of course, is used in finding integrating factors.