Let's suppose that $Y$ is the normal distribution and that $X$ is another random variable whose density function may or may not exist.
Does it follow that $Y+X$ has a density function?
I am reading Resnick's book "A probability path" and this comment was made in passing which confused me. It's an essential step in the proof of the Fourier inversion for characteristic functions.
Yes: as soon as at least one of the random variables $X$ and $Y$ has a density and if $X$ and $Y$ are independent, the sum $X+Y$ has a density.
To understand the magic, assume that $(X,Y)$ is independent, that $Y$ has density $f$ and that $X$ is purely discrete with $P(X=x_n)=p_n$ for every $n$, then $X+Y$ has density $g$ with $$g(x)=\sum_np_n\,f(x-x_n).$$ Without independence, things can go awry: consider $Y$ with a density and $X=-Y$ then $X+Y$ has no density.