Almost surely convergence related with independent suppose that $ x_1 , x_2 $ are independent observation of a random variable x 
prove that $x_1 +x_2$ has the same distribution as  $x  \iff x=0 $ a.s
please please help me for this problem 

If $X_{1}+X_{2}$ has the same distribution as $X_{1}$, then
\begin{align*}
 Var[X_1] 
 &= Var[X_1+X_2] \\
 &= Var[X_1]+Var[X_2]
 & \text{Independence}         
\end{align*}
Conclude that $Var[X_2]=0$. Since the variables are identically distributed,
$X_1=X_2=k$. Since $X_{1}+X_{2}$ has the same distribution as $X_{1}$, $2k=k$, that is $k=0$. 
