For a problem I am currently working on, I would like to calculate the integral over the space $(\{1,2,\dots,n\}^\mathbb{N},\mathcal{F},\nu)$ where $\nu$ is the markov measure.
My problem is that that it doesn't seem possible to work with the traditional form one would use. I have:
$ \int_X f(x) \,d\nu(x) = \sum_{x_0=1}^n\sum_{x_1=1}^n\sum_{x_2=1}^n\dots f((x_0,x_1,x_2,\dots)) \nu(x_0,x_1,x_2,\dots)$
which doesn't really allow much computational flexibility due to the infinite number sums. Are there any useful tricks or general forms which make this more computable?
PS: in my case I have $f(x) = I_{\alpha|\mathcal{F}}(x)$, where $I_{\alpha}(x) = -\sum_{A \in \alpha}1_{A}(x) \log \nu(A)$ and $\alpha$ a partition of $X$.