# Reverse kernel of a Markov kernel with density

Let

• $$(E_i,\mathcal E_i)$$ be a measurable space
• $$\kappa$$ be a Markov kernel with source $$(E_1,\mathcal E_1)$$ and target $$(E_2,\mathcal E_2)$$

Assume $$\kappa$$ has a positive density with respect to a measure $$\mu$$ on $$(E_2,\mathcal E_2)$$, i.e. there is a $$\mathcal E_1\otimes\mathcal E_2$$-measurable $$f:E_1\times E_2\to(0,\infty)$$ with $$\kappa(x,\;\cdot\;)=f(x,\;\cdot\;)\mu\;\;\;\text{for all }x\in E_1.$$ Now, let $$\nu$$ be a probability measure on $$(E_1,\mathcal E_1)$$ and $$\overleftarrow\kappa_\nu(y,\;\cdot\;):=\frac1{c(y)}f(\;\cdot\;,y)\nu\;\;\;\text{for }y\in E_2,$$ where $$c(y):=\int\nu({\rm d}x)f(x,y)$$ (and we assume that $$c(y)<\infty$$) for $$y\in E_2$$.

How can we show that $$\overleftarrow\kappa_\nu$$ is the reverse kernel of $$\kappa$$ with respect to $$\nu$$ (see Definition 2.1.2), i.e.$$^1$$ $$\int\nu({\rm d}x)\int\kappa(x,{\rm d}y)g(x,y)=\int\nu\kappa({\rm d}y)\int\overleftarrow\kappa_\nu(y,{\rm d}x)g(x,y)\tag1$$ for all bounded and $$\mathcal E_1\otimes\mathcal E_2$$-measurable $$g:E_1\times E_2\to\mathbb R$$?

Let $$\pi:E_1\times E_2\to E_2\times E_1\;,\;\;\;(x,y)\mapsto(y,x).$$ It's easy to observe that the left-hand side of $$(1)$$ is equal to$$^2$$ $$\int g\:{\rm d}(\nu\otimes\kappa)\tag2$$ and the right-hand side is equal to $$\int g\circ\pi^{-1}\:{\rm d}(\nu\kappa\otimes\overleftarrow\kappa_\nu).\tag3$$

Now, it's easy to see that$$^3$$ $$\pi_\ast(\nu\otimes\kappa)=\mu\otimes\overleftarrow\kappa_\nu\tag4$$ and $$\nu\otimes\kappa=\left(\pi^{-1}\right)_\ast(\mu\otimes\overleftarrow\kappa_\nu)\tag5.$$

$$(1)$$ is claimed in the linked document below the Definition. Could it be the case that their definition of "reverse kernel" is broken? From a terminological point of view it would make more sense to me if $$\nu\kappa$$ on the right-hand side of $$(1)$$ would be replaced by $$\nu$$.

$$^1$$ $$\nu\kappa$$ denotes the composition of $$\nu$$ and $$\kappa$$.

$$^2$$ $$\nu\otimes\kappa$$ denotes the product of $$\nu$$ and $$\kappa$$.

$$^3$$ $$\pi_\ast(\nu\otimes\kappa)$$ denotes the pushforward measure of $$\pi$$ with respect to $$\nu\otimes\kappa$$.