Given a probability space $(\mathbb{R}^k, \mathcal{B}^k, \mathbb{P}_1)$ and a Markov kernel $\mathbb{P}_{1,2}:\mathbb{R}^k\times\mathcal{B}^l$ on a measure space $(\mathbb{R}^l, \mathcal{B}^l)$. Then there exists a probability measure $\mathbb{P}$ given by $$\mathbb{P}(A)=\int_{\mathbb{R}^k}\left(\int_{\mathbb{R}^l}\mathbb{1}_{A}(\omega_1, \omega_2)\mathbb{P}_{1,2}(\omega_1, d\omega_2)\right)\mathbb{P}_1(d\omega_2),$$ for $A\in \mathcal{B}^k \otimes \mathcal{B}^l=\mathcal{B}^{k+l}$.

If we now introduce random variables $X_1$ and $X_2$ on $(\mathbb{R}^k, \mathcal{B}^k, \mathbb{P}_1)$ and $(\mathbb{R}^l, \mathcal{B}^l)$, respectively, we can form the random variable $(X_1, X_2)$ defined on $(\mathbb{R}^{k+l}, \mathcal{B}^{k+l}, \mathbb{P})$.

1.) So now I want to write down the distribution of the random variable $X_1$, i.e., $\mathbb{P}_{X_1}$. This is a push forward measure. But my question is, a pushforward measure of which measure? I.e., of $\mathbb{P}_1$ or of $\mathbb{P}$?

2.) Can it maybe be discribed in terms of either one? If so how would it look like if it is defined as the push forward measure of $\mathbb{P}$, which takes as an input sets from $\mathcal{B}^{k+l}$, but $X_1$ is only defined on $\mathbb{R}^k$?

3.) More generally, given a random variable $X$ and a random variable $Y$ on two different spaces $(\mathbb{R}^k, \mathcal{B}^k)$ and $(\mathbb{R}^l, \mathcal{B}^l)$, that have a joint distribution $\mathbb{P}_{X, Y}$ on $(\mathbb{R}^{k+l}, \mathcal{B}^{k+l})$. Then of what measure is $\mathbb{P}_X$ a pushforward measure of? And of what measure is the joint distribution a forward measure of? I.e., do we require a measure $\mathbb{P}$ on the product space $(\mathbb{R}^{k+l}, \mathcal{B}^{k+l})$ and are the marginals and the joint distribution push forward measure of that $\mathbb{P}$? If, so, how does it work for $\mathbb{P}_X$ as the dimensions dont match?

Given a probability space $(\mathbb{R}^k, \mathcal{B}^k, \mathbb{P}_1)$ and a Markov kernel $\mathbb{P}_{1,2}:\mathbb{R}^k\times\mathcal{B}^l\rightarrow\mathbb{R}$ on a measure space $(\mathbb{R}^l, \mathcal{B}^l)$. Then there exists a probability measure $\mathbb{P}$ given by $$\mathbb{P}(A)=\int_{\mathbb{R}^k}\left(\int_{\mathbb{R}^l}\mathbb{1}_{A}(\omega_1, \omega_2)\mathbb{P}_{1,2}(\omega_1, d\omega_2)\right)\mathbb{P}_1(d\omega_2),$$ for $A\in \mathcal{B}^k \otimes \mathcal{B}^l=\mathcal{B}^{k+l}$.

If we now introduce random variables $X_1$ and $X_2$ on $(\mathbb{R}^k, \mathcal{B}^k, \mathbb{P}_1)$ and $(\mathbb{R}^l, \mathcal{B}^l)$, respectively, we can form the random variable $(X_1, X_2)$ defined on $(\mathbb{R}^{k+l}, \mathcal{B}^{k+l}, \mathbb{P})$.

1.) So now I want to write down the distribution of the random variable $X_1$, i.e., $\mathbb{P}_{X_1}$. This is a push forward measure. But my question is, a pushforward measure of which measure? I.e., of $\mathbb{P}_1$ or of $\mathbb{P}$?

2.) Can it maybe be discribed in terms of either one? If so how would it look like if it is defined as the push forward measure of $\mathbb{P}$, which takes as an input sets from $\mathcal{B}^{k+l}$, but $X_1$ is only defined on $\mathbb{R}^k$?

3.) More generally, given a random variable $X$ and a random variable $Y$ on two different spaces $(\mathbb{R}^k, \mathcal{B}^k)$ and $(\mathbb{R}^l, \mathcal{B}^l)$, that have a joint distribution $\mathbb{P}_{X, Y}$ on $(\mathbb{R}^{k+l}, \mathcal{B}^{k+l})$. Then of what measure is $\mathbb{P}_X$ a pushforward measure of? And of what measure is the joint distribution a forward measure of? I.e., do we require a measure $\mathbb{P}$ on the product space $(\mathbb{R}^{k+l}, \mathcal{B}^{k+l})$ and are the marginals and the joint distribution push forward measure of that $\mathbb{P}$? If, so, how does it work for $\mathbb{P}_X$ as the dimensions dont match?