Disintegration Theorem

Michikazu Hirata 📧

November 2, 2023

This is a development version of this entry. It might change over time and is not stable. Please refer to release versions for citations.


We formalize mixture and disintegraion of measures. This entry is a formalization of Chapter 14.D of the book by Baccelli et.al. The main result is the disintegration theorem: let $(X,\Sigma_X)$ be a measurable space, $(Y,\Sigma_Y)$ be a standard Borel space, $\nu$ be a $\sigma$-finite measure on $X \times Y$, and $\nu_X$ be the marginal measure on $X$ defined by $\nu_X(A) = \nu(A\times Y)$. Assume that $\nu_X$ is $\sigma$-finite, then there exists a probability kernel $\kappa$ from $X$ to $Y$ such that \[ \nu(A\times B) = \int_A \kappa_x (B) \nu_X(\mathrm{d}x), \: A\in\Sigma_X, B\in\Sigma_Y.\] Such a probability kernel is unique $\nu_X$-almost everywhere.


BSD License


Session Disintegration