This is an exercise in Jacod and Protter's Probability Essentials:
Let $X$ and $Y$ be independent random variables and $P(X+Y=\alpha)=1$ where $\alpha\in{\Bbb R}$ is some constant. Show that both $X$ and $Y$ are constant random variables.
What I think is that one might use Borel-Cantelli theorem here. Since $ \bigcup_{i=0}^{\infty}\{X=i,Y=\alpha-i\}\subset\{X+Y=\alpha\}=\bigcup_{\beta\in{\Bbb R}}\{X=\beta,Y=\alpha-\beta\}, $ we have $ \begin{align} P\bigg(\bigcup_{i=0}^{\infty}\{X=i,Y=\alpha-i\}\bigg)=\sum_{i=0}^{\infty}P(X=i,Y=\alpha-i)<\infty \end{align} $ But this seems to give nothing. Also, I'm surprised about the result is that $X=\gamma$ for some constant $\gamma$ instead of $P(X=\gamma)=1$. Any idea about how I can go on?