What does this expression mean (probability distribution for nature) - [game theory]

Question

I am working with Mailath's GT book and I stuck at this point.

"A finite extensive form game $\Gamma$ consists of a finite extensive form with:
...
A probability distribution for nature: $\rho: X_0 \rightarrow U_{t\in X_0}\Delta(A(t))$ such that $\rho(t) \in \Delta(A(t))$

Where A(t) is the set of actions available at node t, and $X_0$ is the set of nodes where nature plays"

Here are the things I don't know:

1. What does this $\Delta$ mean?
2. What does this probability $\rho(t)$ mean? Does it mean "the probability of nature taking each action in a given node t"?

Answer 1

1. It denotes the N-1 dimensional simplex. In this case it is just a formal way of describing the set of discrete probability distributions over a set of actions. The statement is that for each $X_{0}$ $\rho$ assigns a probability distribution over the actions.
2. Yes, it is precisely that.

Please let me know if you need further explanation.