Joint pdf problem on a general space I came across this question in a book and having trouble understanding it. 
Suppose $(X,Y)$ are continuous random vector with joint pdf $f(x,y)$ and support $\mathcal{X} \times \mathcal{Y}$. In particular suppose that the marginals $f_X(x)$ and $f_Y(y)$ have support $\mathcal{X}$ and $\mathcal{Y}$ respectively. I need to verify that $k(x,y|x^\prime, y^\prime) = f(x|y^\prime)f(y|x)$ is a density function.
I think this problem has to do with applying Fubini theorem but what confuses me are $x^\prime$ and $y^\prime$. There is no mention to what they mean.Should I consider them to be other random variables distinct from $x$ and $y$ but with supports in $\mathcal{X}$ and $\mathcal{Y}$? I am not sure of how to understand this problem. Any help would be appreciated. 
 A: Apart from an abuse of notation ($f$ having multiple meanings), there is no true difficulty in defining
$$
k(x,y|x^\prime, y^\prime) = f(x|y^\prime)f(y|x)
$$
as a density on the pair $(X,Y)$ since 


*

*The conditional density of $Y$ given $X$ is $f(y|x)$, which integrates to one;

*The marginal density of $X$ is $f(x|y')$, which also integrates to one.


What may sound confusing is the parametrisation in $(x',y')$ but this simply indicates that the distribution does not depend on $x'$. A standard occurrence of this setting is in Gibbs sampling or slice sampling (which is possibly where you found this description): when using two full conditionals for the Markov transition, one simulates [at iteration $t$]


*

*$X_{t+1}|X_t,Y_t\sim f(x|y_t)$

*$Y_{t+1}|X_t,Y_t,X_{t+1} \sim f(y|x_{t+1})$


or $$(X_{t+1},Y_{t+1})|(X_{t},Y_{t})\sim k(x,y|x_t,y_t)$$
and the move does not directly depend on the value of $X_t$.
Note that this is not incompatible with 
$$(X_{t+1},Y_{t+1})\sim f(x,y)$$
in the stationary regime of the Markov chain, since $f(x,y)$ is the stationary density while $k(x,y|x_t,y_t)$ is the conditional density at iteration $t+1$.
