Let $X$ be a standard Borel space, and $P(X)$ be space of Borel probability measures on $X$. It is also a standard Borel space if endowed with the topology of weak convergence, so we can integrate over it. Consider a functional $f:P(X) \to \Bbb R$; we say that $f$ is good whenever $$ f\left(\int_{P(X)}p\;\nu(dp)\right) = \int_{P(X)}f(p)\nu(dp) \tag{1} $$ for each $\nu\in P(P(X))$. For example, if we'd consider only discrete measures $\nu$, then $(1)$ would define exactly linear functionals. My question are the following:
Are there any linear functionals $f$ that are not good?
On $P(X)$ any good functional is exactly of the form $f(p) = \int_X g \,dp$. If instead of $P(X)$ we would consider a general topological vector space $V$ and $\nu$ in $(1)$ would be any Borel measure on $V$, is there any special term for functions satisfying $(1)$ in that setting?
I have asked a related question on MSE but have not received a definite answer.
