Consider the spaces
- $\mathcal{M}$ the set of all finite, signed measures on $(\mathbb{R},\mathcal{B})$
- $\Phi$ the set of all right-continuous functions $F:\mathbb{R}\to\mathbb{R}$ with bounded variation and $\lim_{x\to-\infty}F(x)=0$
Then the projection $$\mu\mapsto F_\mu(x):= \mu(]-\infty,x])\,, x\in\mathbb{R}$$ is an isomorphism. Further, we can equip $\Phi$ and $\mathcal{M}$ with the total variation norms (for functions and measures respectively). Then, because $F\cdot G\in\Phi$ and because both spaces are isomorphic, there has to be an equivalent operation on $\mathcal{M}$ to the multiplication of functions in $\Phi$.
Does someone know how this operation is defined on $\mathcal{M}$?