Thank you very much for your support, but yesterday i found the answer in my stochastic I scriptum...if someone is interested in:
Let x, y be independent rvs with dfs F and G then holds for the distribuntion of the sum x+y:
H(a)=P(x+y<=a)=int F(a-v)dG(v).
That's almost all clear but in my book i read that F(s) wrt F(s) is something like F(s)*F(s) which stands here for the convolution...
And i have a x in the itegrand as well as in the limit.
As I said before: int(0 to x) 1-F(x-t) wrt F(t), sorry so you are write, it is F(t) not F(x)!
It helps, but that mean to multiply the integrand by the density f=dF/dx and then integrate over dx, so like how to derive the expectation...
but another problem is, that I do not know the density and need to have a result within F(x)