Given a measure space $\mathcal M$, I am wondering what kind of measure space $\mathcal T(\mathcal M)$ one could associate to the set of binary trees with elements from $\mathcal M$ at each node.


The kind of trees I mean can probably best be described in functional programming syntax:



```
datatype Tree(a) = Leaf | Node(Tree(a), a, Tree(a))

```

In particular, I wonder what the $\sigma$-algebra of that $\mathcal T(\mathcal M)$ would look like, and how this generalises to other algebraic datatypes.



Measure space for trees and other algebraic datatypes
The tree structures associated with a partition of the sample space $\mathcal{X}$ is usually discussed along with a Beta process(or in computer engineers' world they refer it as "stick-breaking process"). In statistics, this is very useful in nonparametric estimations. When you try to designate a random measure $m\in\mathcal{M}=\mathcal{M(\mathcal{X})}$ for your model. Since $m$ is random, we can also talk about $m(\omega)$ and the stochastic process that generates $m$ as a sample path $m(\omega),\omega\in \mathcal{X}^{\prod}$. Usually the sample path of the Beta process will give a tree-like structure on the product sample space $\mathcal{X}^{\prod}$. The sigma algebra on this product space correponds to the filtration of the Beta process. That is how the $\sigma$-algebra on $T(\mathcal{M})$ looks like.


For technical details, please refer to Chap3-4 of 



> 
> Ghosh, Jayanta K., R. V. J. K. Ghosh, and R. V. Ramamoorthi. Bayesian
>  nonparametrics. 2003.
> 
> 
> 


Another useful explnatory paper is <http://projecteuclid.org/download/pdfview_1/euclid.cbms/1362163749> 
which is part of the book <https://projecteuclid.org/euclid.cbms/1362163742>



> 
> Müller, Peter, and Abel Rodriguez. Nonparametric bayesian inference.
>  Institute of Mathematical Statistics, 2013.
> 
> 
> 


