Is the disintegration of two borelian probabilities measures is associative ? It means if $\mu = \mu_{y}^{1} \oplus h_{\#}^{1}\mu$ and $ h_{\#}^{1}\mu = \mu_{y}^{2} \oplus h_{\#}^{2} h_{\#}^{1}\mu$. Then do we have $$ \mu = \mu_{y}^{1} \oplus \mu_{y}^{2} \oplus (h_{2} \circ h_{1})_{\#} \mu $$ Where $\mu$ is a borelian probability over $\mathbb{R}^{d}$ and I note $\mu = \mu_{y} \oplus h_{\#}\mu$ the disintegration of $\mu$ according to $h_{\#} \mu$ that gives the kernel $\mu_{y}$ with $y \in \mathbb{R}^{d}$.