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

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