What is the relationship between the Law of Total Expectation and the Disintegration Theorem?
I am interested both in the general case $\mathbb{E}[\mathbb{E}[X \mid \mathcal{G}] \mid ]\mathcal{H}] = \mathbb{E}[X \mid \mathcal{H}]$ and in the specific and more common case $\mathbb{E}[\mathbb{E}[X\mid Y]] = \mathbb{E}[X]$. Somehow TOWER must be a particular disintegration right? Possibly a disintegration into regular conditional probabilities?