# What is the set of all probability distributions?

I have a little problem which is in part due to my lack in english skills. I am currently reading a book and I dont understand the following chapter (RW stands for random walk): part1
part2
The problem is i dont understand what the author means with $\lambda \in P(\mathbb{R})$ (the set of all probability distributions). What is a probability distribution? THe wikipedia article is not quite clear. Im not sure if $\lambda$ is a distribution function or a probability measure on R. Now up to this point The F and its n-th fold have been distribution functions. But the proof that a standard model always exists would imply that the F's and the $\lambda$ are probability measures now.

• In your link, both $\lambda$ and $F$ are probability measures. – fourierwho Dec 23 '17 at 22:36
A probability distribution is a function F(x) defined on the real line with the following properties: $F(x)$ is monotone increasing, $\lim_{x \to -\infty}F(x)=0,\ \lim_{x \to \infty}F(x)=1$.