Let $(E,\mathcal{E})$ be a measure space and let $(m_{n})_{n\in\mathbb{N}} \subset \mathcal{M}_\sigma(E,\mathcal{E})$ be a sequence of $\sigma$-finite measures. I will put my questions and state below what I found so far. I would be glad to obtain some references.

- Is there a notion of weak convergence in this general setting?
- Assume $E = \bigcup_{i\in \mathbb{N}} E_i$ for some $E_i \in \mathcal{E}$ and $E_i \subset E_{i+1}$ for $i \in \mathbb{N}$. Suppose all $m_n$ are finite on each of the $E_i$: $m_n(E_i) < \infty, \, i,n \in \mathbb{N}$. Is there a notion of weak convergence?
- In the setting of 2. Is there a generalization of Prohorov's theorem if we assume $E$ to be Polish and equipped with its Borel $\sigma$-field? Reference?

ad 2: I have seen some work by Matyas Barczy and Gyula Pap on a similar question (see http://www.inf.unideb.hu/valseg/dolgozok/barczy/barczy_pap3.pdf). There $E$ is equipped with a metric and has a distinguished point $x_0 \in E$. There $E_i = E \setminus B(x_0, i^{-1})$, i.e. they remove small balls from $E$.

Using their definition it seems reasonable to define $m_n \Rightarrow m$ iff $m_n|_{E_i} \Rightarrow m|_{E_i}$ as $n\to \infty$ for any $i \in \mathbb{N}$. This uniquely defines a $\sigma$-finite measure $m \in \mathcal{M}_\sigma(E,\mathcal{E})$ using an extension theorem for measures. (I write $m|_{E_i}$ for the restriction of $m$ to $E_i$: $m|_{E_i}(A):=m(A\cap E_i)$).

ad 3: If one requires tightness of finite measures for $(m_n|_{E_i})_{n\in \mathbb{N}}$ for any $i \in \mathbb{N}$ then one can apply the standard Prohorov theorem restricted to $E_i$ and do a diagonalization argument to obtain a limit.

vague convergence. A good description of the concept can be found in Appendix 2.6 ofAn Introduction to the Theory of Point Processes: Volume I, by D.J. Daley and D. Vere-Jones. My main interest was the situation $E = \mathbb{R}^d \setminus \{0\}$ so I am happy with this source. $\endgroup$