Technically (jensen equation) as f(x+y)=F(x)+F(y) when F(0)=0 and I can see how given the unit f(1)=1, one could effectively capture (numerically) powers of three , 7 etc, in a probability setting, and how this can be generalized, but I am not convinced, that the two equations literally the same in the finite case (before continuity is applied, just with F(0)=0 and F(1)=1 if need be in a probability setting. Can it express this in a functional manner for all rational sigma F(sigma(x+y))=F(sigma \timesx)+f(sigma\timesy)
Whilst cauchy's equation and jensens equation could be considered functionally equivalent given continuity and f(0)=0 etc), are they identical in the dense infinite (but rational) cases at least when f(0)=0.
For example can it precisely but can it precisely express F(x+y +z_=F(x)+F(y)+F(z) for all rationals; and does it entail rational homogeneity before continuity as applied as does cauchy's equation.
For all real numbers F(x+y)=F(x)+F(y) and for all real numbers where for all rat sigma, F(sigma\times A)= sigma times F(A)in this form or in the additive jensen form athough it appears to be satisfies conditionally for all x though, and the numerically results bear it out.