I'm familiar with this definition of convolution :
$$f*g(x)=\int f(x-y)g(y) dy$$
But anyone help me see the link between this one and the two definitions hereafter? :
Definition 1 : $$f*g(A)=\int_Xf(A-x)dg(x)$$
where $A-x=\{a-x|a\in A\}$
Definition 2: $$f*g(A)=\int\int 1_A (x+y)dg(x)df(y)$$
thank you
Your definition 1 makes sense for measures: $$\mu*\nu(E) = \int\mu(E-y)d\nu(y).$$ For each $y$, $E-y$ is a set, $\mu(E-y)$ is a number and $y\mapsto\mu(E-y)$ is a function that can be integrated.
For your definition 2, think in the previous formula in the case $$\mu(A) = \int_A f(x)\,dx,\qquad\nu(A) = \int_A g(x)\,dx.$$