Let $X$ and $Y$ be two independent integer-valued random variables, with distribution functions $m_1(x)$ and $m_2(x)$ respectively. Then the convolution of $m_1(x)$ and $m_2(x)$ is the distribution function $m_3 = m_1 * m_2$ given by:
$m_{3}(j)=\sum_{k}m_{1}(j-k)m_{2}(k)$
Until this i have understood everything.
Now, what are the detailed calculations ('from discrete sum to integral') that allow us to go from the equation written above to the continuos one ?
$(f*g)(z) = \int_{-\infty }^{+\infty } f(z-y)g(y)dy$
In what follows Lebesgue measures are involved. If counting measures are involved then we get the same result for PMF's.
There is - as far as I know - no route that deduces the case of Lebesgue measures from the case of counting measures.
In other words: there is no road "from discrete to continuous" concerning densities.
Let $f$ denote the PDF of random variable $X$ and let $g$ the PDF of random variable $Y$ and let $X,Y$ be independent.
Substitution of $z=x+y$ in: $$\int1_{\left(-\infty,t\right]}\left(z\right)f\left(z-y\right)g\left(y\right)dz$$ gives: $$\int1_{\left(-\infty,t\right]}\left(x+y\right)f\left(x\right)g\left(y\right)dx$$
Further: $$\int\int1_{\left(-\infty,t\right]}\left(z\right)f\left(z-y\right)g\left(y\right)dzdy=\int\int1_{\left(-\infty,t\right]}\left(z\right)f\left(z-y\right)g\left(y\right)dydz=$$$$\int_{-\infty}^{t}\int_{-\infty}^{\infty}f\left(z-y\right)g\left(y\right)dydz$$
and: $$\int\int1_{\left(-\infty,t\right]}\left(x+y\right)f\left(x\right)g\left(y\right)dxdy=P\left(X+Y\leq t\right)$$
So: $$P\left(X+Y\leq t\right)=\int_{-\infty}^{t}\int_{-\infty}^{\infty}f\left(z-y\right)g\left(y\right)dydz$$ showing that the function $z\mapsto\int_{-\infty}^{\infty}f\left(z-y\right)g\left(y\right)dy$ serves as PDF of $X+Y$.