Cumulative distribution over a set If $F$ is an arbitrary cumulative distribution function of a random variable $X$, and $S$ a subset of the support of $X$, what does $F(S)$ mean? Is it $\int_S dF(x)$ ? 
In particular, if X is defined on $[0, 1]$, what does it mean to say that there is a set $S$ such that $F(S) = 1$?
This is probably a trivial question, but I have never seen this anywhere, except for old statistical learning papers (e.g. [1]), where it is used without a definition.
[1] Gordon, L., Olshen, R. A. (1978). Asymptotically efficient solutions to the
classification problem. Ann. Statist. 6 5 15-533.
 A: I think you are confusing the cumulative distribution function (CDF) and the associated probability measure induced by that function.  For a real scalar random variable, the CDF is a function $F:\mathbb{R} \rightarrow \mathbb{R}$, so it would not take a set $\mathcal{S} \subseteq \mathbb{R}$ as an input at all.$^\dagger$
The probability measure $\mathbb{P}$ and the corresponding CDF $F$ are related by the fact that:
$$\mathbb{P}(( -\infty, x ]) = F(x)
\quad \quad \quad
\text{for all } x \in \mathbb{R}.$$
For a measureable set $\mathcal{S} \subseteq \mathbb{R}$ you have:
$$\mathbb{P}(\mathcal{S}) = \int \limits_\mathcal{S} dF(x).$$
If $\mathbb{P}(\mathcal{S}) = 1$ then it means that the random variable $X$ is somewhere in the set $\mathcal{S}$ with probability one (i.e., almost surely).  

$^\dagger$ As noted in the answer by spiritlevel, the notation $F(\mathcal{S})$ could arguably be used to refer to the image of $F$ on the set $\mathcal{S}$.  However, even if that were the intention, you would not get $F(\mathcal{S})=1$.  There is rarely any good reason to take the image of a CDF on a set.
A: For a set $A$ in the domain of a function $f$, I've seen the following notation used quite a bit:
$$
f(A) = \{ f(a) : a\in A \}
$$
