Assume that we have two independent random variables $X_1$ and $X_2$ with distribution functions $F_1$ and $F_2$ respectively. Then Theorem 6.1.1. on Kai Lai Chung's "A course in probability theory" states that:
The sum $X_1+X_2$ has distribution function $F_1*F_2$.
On this set of notes the author states that actually the distribution of the sum is NOT the convolution of the distribution functions, but rather the convolution of the $F_1$ and the density $f_2$.
For what know I about the definition of convolution
$\int F_1(x-x_1) dF_2(x_2)$ looks more like the convolution $F_1*f_2$, I think that this is more a problem of notation but still I am curious.
EDIT: Another example on which the convolution is taken between the distribution functions is "Probability Theory and examples" by Rick Durret.
This question hinges on the definition of "convolution" and of the $*$ symbol. A glance at the wikipedia article gives one formulation for the convolution of two probability measures: it is the probability measure of the sum of two independent random variables, equivalent to $(\mu*\nu)(A)=P(X+Y\in A)$ if $X\sim\mu$ and $Y\sim\nu$. From this one has $$F_{X+Y}(t)=P(X+Y\le t)=E[(P(X+Y\le t|Y)]=E[ P(X\le t-Y|Y)] = E[ F_X(t-Y)|Y] = \int_{\mathbb{R}}F_X(t-y) F_Y(dy) = \int_{\mathbb{R}}F_X(t-y) dF_Y(y),$$ which one might as well take to be the definition of the convolution of distributions functions of probability measures, regardless of whether the random variables $X$ and $Y$ are continuous or discrete, with densities or without.
The notes the OP cites are not reliable on this issue: they adopt the dumbing-down restriction to probability laws with density functions. I do not know what Ross says, but I think Chung (the most trustworthy of the lot) does not make mistakes like this.