Let $H:L^2[0,1]\rightarrow \mathbb{R}$ satisfy
$$H(f*g)=H(f)+H(g).$$
**Question**:Is there a characterization of all such functionals $H$? 

**Related questions**:Can it be extended to measures? If so, is it enough to know that the equation holds for discrete measures?

PS: I earlier asked [this related question](https://mathoverflow.net/questions/144593/characterization-of-the-laplace-transform)