I would like to show:

> 
> Let $X,Y$ be independent RVs.
> If there exists $c \in \mathbb R $ s.t. $P(X+Y=c)=1$, then $X,Y$ are constants a.s..
> 
> 
> 

What I tried:

Since X,Y are indep,$$1=P(X+Y=c)=\int 1\_{x+y=c} \, d \mu \_x \, d \mu\_y = \int P(X=c-y) \, d \mu\_y$$ but it gets nowhere from here.