I am trying to understand the arguments in a book I am reading.
Consider the probability space $\left( X, \mathcal{B} \right)$ and let $\mathcal{P}$ be the set of probability measures on it. Let $\mathcal{C}$ be the $\sigma$-algebra generated by the sets $A_{B, t} = \left\{ P \in \mathcal{P}: P \left( B \right) \leqslant t \right\}$ where $B \in \mathcal{B}$ and $t \in \left[ 0, 1 \right]$.
- How does one prove that $\mathcal{C}$ is the smallest $\sigma$-algebra making the functions $g_B$ from $\mathcal{P}$ into $\mathbb{R}$ defined by $g_B \left( P \right) = P \left( B \right)$ measurable?
- I am looking also for specific counterexamples of functions not measurable in this context to help me understand it more.
- It is also indicated in the book that there is some link to the topology of pointwise convergence. What is that link?