Generalization of a product measure Let $(X,\mathfrak B(X))$ and $(Y,\mathfrak B(Y))$ be measurable spaces and further let $\mu$ be a measure on $\mathfrak B(X)$ and let $K$ be a kernel, i.e. for any $x\in X$ we have $K_x$ is a measure on $\mathfrak B(Y)$ and the map $x\mapsto K_x(B)$ is $\mathfrak B(X)$-measurable for any $B\in \mathfrak B(Y)$. Let us further assume that $\mu$ is finite and measures $K_x$ are uniformly bounded. Then there exists a unique measure $P$ on $\mathfrak B(X)\otimes \mathfrak B(Y)$ such that
$$
  P(A\times B) = \int_AK_x(B)\mu(\mathrm dx)
$$
for any measurable rectangle $A\times B$. Thus, $P$ can be considered as a certain product $\mu\otimes_{?}K$ and in case $K_x\equiv\nu$ we have $P = \mu\otimes \nu$ is just a product measure. We know that product measures are not enough to describe all possible measure over the product space, whereas the construction above is thanks to the disintegration theorems. I wonder what are the nice sources for the properties of the construction $\mu\otimes_{?}K$ - e.g. does it have its own name.
 A: The following is in terms of probability kernels, which I'm most familiar with, but much of it generalizes. If $(X,\mathcal{X})$, $(Y,\mathcal{Y})$, and $(Z,\mathcal{Z})$ are measurable spaces and $\kappa_1:X\times\mathcal{Y}\to[0,1]$ and $\kappa_2:Y\times\mathcal{Z}\to[0,1]$ are probability kernels, we can define a composition $\kappa_2\circ\kappa_1:X\times\mathcal{Z}\to [0,1]$ by $$\kappa_2\circ\kappa_1(x,B)=\int\kappa_2(y,B)~d\kappa_1(x,d(y)).$$
One can show that this composition is associative. Measurable spaces and probability kernels form a category. This category was called the category of probabilistic mappings by Lawvere in unpublished notes from 1962. The same category was named the category of statistical decision functions in the book Statistical Decision Rules and Optimal Inference by Cencov (based on his dissertation). A nice overview of material on this category can be found in the introduction of a paper by Culbertson and Sturtz.
If $f:X\to Y$ is a measurable mapping, there is a corresponding kernel $\kappa_f:X\times\mathcal{Y}\to[0,1]$ given by $$\kappa_f(x,B) =
\begin{cases}
1 & \text{if } f(x)\in B\\
0 & \text{if } f(x)\notin B.
\end{cases}$$ 
This way, we can view measurable mappings as special kernels. If $(X_i,\mathcal{X}_i)$ and $(Y_i,\mathcal{Y}_i)$ are families of measurable spaces and $\kappa_i:X_i\times\mathcal{Y}_i\to[0,1]$ is a kernel for all $i$, then there is a product-kernel $$\kappa_\times:\prod_i X_i\times\otimes_i\mathcal{Y}_i\to [0,1]$$ such that $\kappa_\times(x,\cdot)=\otimes_i\kappa_i(x_i,\cdot)$. 
We can also treat a probability measure as a constant-valued kernel. If we combine these ideas, we can take a probability measure (seen as a kernel) $\mu$ on $(X,\mathcal{X})$ and a kernel $\kappa:X\times\mathcal{Y}\to [0,1]$ to get a natural distribution on $\mathcal{X}\otimes\mathcal{Y}$. We let $p:X\to X\times X$ be given by $x\mapsto (x,x)$. Let $\kappa_p:X\to\mathcal{X}\otimes \mathcal{X}$ be the corresponding kernel. Let $\kappa_1:X\times\mathcal{X}$ be the kernel corresponding to the identity on $X$. We let $\kappa_\times$ be the product kernel of $\kappa_1$ and $\kappa$. Then $\kappa_\times\circ\kappa_p\circ\mu$ is essentially the natural distribution on $\mathcal{X}\otimes\mathcal{Y}$ such that $\mu$ is the marginal on $X$ and $\kappa$ gives the conditional probabilities on $Y$ given $x$. So the generalized product measure is really a special form of composing kernels.
A: There are lecture notes from Erwin Bolthausen for a course in stochastic processes in discrete time, albeit in German: www.math.uzh.ch/index.php?file&key1=11678
On page 17 he introduces Markov kernels and on page 19 the construction from the OP is defined and called »semidirektes Produktmass«, which could be translated as »semidirect product measure«. Shortly after that a sort of wedge product between kernels is introduced. All this notation is needed for his proof of the Ionescu-Tulcea Theorem.
