I'm trying to prove the next:
Let $M(\mathcal{B}_{\mathbb{R}})$ be the space of all finite measures over $(\mathbb{R},\mathcal{B}_{\mathbb{R}}).$ Let $\lambda$ be the Lebesgue measure over real numbers, and $\mu,\nu\in M(\mathcal{B}_{\mathbb{R}}).$ If $\mu<<\lambda$ then $\mu*\nu<<\lambda$ and $$\frac{d(\mu*\nu)}{d(\lambda)}=\frac{d\mu}{d\lambda}*\frac{d\nu}{d\lambda}.$$
Here $\mu*\nu$ denotes convolution between both measures.
I've proved absolute continuity $\mu*\nu<<\lambda$ but I am having trouble with the equality; I don't feel comfortable with my proof:
Let $f,g$ be the Radon-Nikodym derivatives of $\mu$ and $\nu,$ respectively. For each $E\in\mathcal{B}_{\mathbb{R}}$ we have $$\mu*\nu(E)=\int_{\mathbb{R}^2} 1_E(x+y)d\mu(x)d\nu(y)=\int_{\mathbb{R}^2} 1_E(x+y)f(x)d\lambda(x)g(y)d\lambda(y).$$
Utilizing variable change $t=x+y$ and the invariance of Lebesgue measure it follows:
$\begin{eqnarray} \mu*\nu(E)&=&\int\int_{\mathbb{R}^2}1_E(t)f(t-y)g(y)d\lambda(t)d\lambda(y) = \int_{\mathbb{R}}1_E(t)\int_{\mathbb{R}}f(t-y)g(y)d\lambda(y)d\lambda(t)\\ &=& \int_{\mathbb{R}}1_E(t)f*g(t)d\lambda(t)=\int_{E}f*g(t)d\lambda(t) \end{eqnarray}$
Then we have the desired result. However I think I am manipulating "algebraically" notation of Radon-Nikodym derivative which is wrong.
How Could we formalize the proof?
Any kind of help is thanked in advanced.
All that you have written (except perhaps a typo in the last display with a triple integral, should be double) seems correct as long as $\nu<<\lambda$ too. Indeed, if so, $d\mu=fd\lambda, d\nu=g d\lambda$ and your manipulations are justified. You can also prove the result using the dual space of $L^1(\lambda)$ functions and your arguments; see the proof of Proposition 8.48(d) in Folland (1999), Real Analysis, 2nd Ed.
As you can find there a few lines after, $\nu<<\lambda$ is not needed and the conclusion (by your arguments too) is that \begin{equation} \frac{d (\mu\ast\nu)}{d\lambda}=\frac{d \mu}{d\lambda}\ast\nu. \end{equation} Note that the result is true for general Euclidean spaces, and that the invariance under shifts of Lebesgue measure is fundamental, i.e., the result is not true for general dominating measures.