can we always decompose a joint probability distribution into the product of a conditional distribution and a marginal distribution?

Question

Let $P$ be an underlying probability measure and $X_1,X_2$ be two random variables. Denote by $P^{(X_1,X_2)}$ the joint distribution of $(X_1,X_2)$ under $P$, $P^{X1}$ the marginal distribution of $X_1$ and $P^{X_2 | X_1}$ the conditional distribution of $X_2$ conditionally to $X_1$. Can we always write $\mathrm{d}P^{X_2 | X_1}\mathrm{d}P^{X1}=\mathrm{d}P^{(X_1,X_2)}$ ?