I would like to know, is it possible that a set of measures can be viewed as a vector space over field $\mathbb{R}$? If yes, under what condition it is a vector space? How to define addition and scalar multiplication?
For example, suppose that $\Omega$ is a finite set. Is $\Delta(\Omega)$ (set of all probability measures on $\Omega$) a vector space? How about $\Delta(\Delta\Omega)$?
If anyone has any clue, please let me know. Thank you.
Best Answer
So, there are actually two different questions asked.
No, measures do not form a vector space (as already pointed out), but they form a module over a rig with scalar multiplication given by Lebesgue integration of measurable nonnegative functions.
Also, more simply: they form a module over the rig of nonnegative reals in the obvious way.
However the story for probability measures is a bit different.
The right structure to put on the set of probability measures is arguably that of a convex space as opposed to a module or vector space structure. This is because $\Delta(\Omega)$ is a convex subset of the $\mathbb R$-vector space of all signed measures on $\Omega$.
The reason my previous answer doesn't work is of course that given $\mu,\nu\in \Delta(\Omega)$ we have $(\mu + \nu)(\Omega) = 2 \neq 1$, so certainly $\Delta(\Omega)$ is not closed under $+$; nor is it closed under scalar multiplication since $c\mu (\Omega) = 1$ if and only, if $c = 1$.
One way around these issues is talking about convex combinations of probability measures instead, hence the need for convex spaces.
Convex spaces have been rediscovered many times and never quite became famous enough to be well-known. The paper by Fritz is a good introduction to the topic.