Finding two functions (density) $g,f$ satisfying some conditions Is there a clever way to find two density functions, $f$ and $g$, that satisfy the following conditions?
$$\begin{align*}
\int_{\infty}^{m}\int_{-\infty}^{\infty}f(w)f(w+z)\,dw\,dz&=\int_{\infty}^{m}\int_{-\infty}^{\infty}g(w)g(w+z)\,dw\,dz\\ 
\int_{\infty}^{m+2}\int_{-\infty}^{\infty}f(w)f(w+z)\,dw\,dz&=\int_{\infty}^{m+1}\int_{-\infty}^{\infty}g(w)g(w+z)\,dw\,dz\\
 \int_{-\infty}^{\infty}f(w)\,dw&=\int_{-\infty}^{\infty}g(w)\,dw=M\\

\end{align*}$$
where $f\gt 0$ and $g\gt 0$ almost everywhere?
for $m\in (-\delta,\delta)$ and $\delta$ is some small number.
My main intent is to come up with two i.i.d. random variable, $X'$ and $X''$ and $Y$ and $Y''$, such that $\operatorname{\mathbb{Pr}}(m> Y'-Y'')=\operatorname{\mathbb{Pr}}(m>X'-X'')$ for $m \in (-b,b)$ for some $b$ small enough, while $\operatorname{\mathbb{Pr}}(m+2> Y'-Y'')=\operatorname{\mathbb{Pr}}(m+1> X'-X'')$.Is this possible?
Thanks so much in advance for your much appreciated help.

Why don't you see if you can work with characteristic function. This is almost related to divisibility, in which case one need only to examine characteristic functions with roots.
Look into Characteristic Functions by Eugene Lukacs.
