So...
$C_0(1)_1=\{0,1,2,3,\dots,\omega,\omega_1\} \cup \{\omega+1,\dots,\omega_1+1,\dots,2\omega_1,\dots,\omega_1^{2},\dots,\omega_1^{ \omega_1},\dots,\psi_{\omega_1}(0),\dots,\psi_{\omega_1}(1)\}$
hence
$C_0(\alpha)_1=\{0,1,\dots,\omega,\dots,\omega+1,\dots,\omega_1+1,\dots,2\omega_1,\dots,\omega_1^{2},\dots,\omega_1^{ \omega_1},\dots,\psi_{\omega_1}(0),\dots,\psi_{\omega_1}(\eta <\alpha)\}$
If $X$ and $Y$ have joint density function
$$
f(x,y)=\begin{cases}
\frac{1}{2}(x+y)e^{-x-y}&\text{if }x,y>0,\\
0&\text{otherwise,}
\end{cases}
$$
find the density function of $X$+$Y$.
I did the following:
$$
\begin{aligned}
f_Z(z)&=\frac{\partial}{\partial z}\frac{1}{2}\int_0^z\int_0^{z-...
