LAUSR

Let $$(S,+)$$ ( S , + ) be an abelian semigroup, let $$\sigma $$ σ be an involution of S ,  let X be a linear space over the field $${\mathbb {K}}\in \{{\mathbb {R}},{\mathbb {C}}\}$$ K ∈ { R , C } and let $$\mu $$ μ , $$\nu $$ ν be linear combinations of Dirac measures. In the present paper, we find the general solution $$f:S\rightarrow X$$ f : S → X of the following functional equation $$\begin{aligned} \int _{S}f(x+y+t)d\mu (t)+\int _{S}f(x+\sigma (y)+t)d\nu (t)=f(x)+f(y), \ \ \ x,y \in S, \end{aligned}$$ ∫ S f ( x + y + t ) d μ ( t ) + ∫ S f ( x + σ ( y ) + t ) d ν ( t ) = f ( x ) + f ( y ) , x , y ∈ S , in terms of additive and bi-additive maps. Many consequences of this result are presented.