Let $X $ and $Y $ be random variables mapping $(\Omega,\mathcal {H },\mathbb{P } ) $ into $(D, \mathcal {D } )$ and $(E,\mathcal {E } )$ respectively. And suppose that their joint distribution $\pi $ is such that for $A \in \mathcal {D } $, $B \in \mathcal {E } $ $$\pi (A \times B )=\int _A\mu (dx )K(x,B )$$ and for any $\mathcal {D } \otimes \mathcal {E } $-measurable function $f $ $$\int _{D \times E } \pi (dx,dy )f(x,y )= \int _D \int _E K (x,dy )f (x,y )$$ where $\mu $ is a measure on $(D, \mathcal {D } )$ and $K $ is a transition kernel from $(D, \mathcal {D } )$ into $(E,\mathcal {E } )$.
Show that the kernel $L $ defined by $L(\omega , B )=K(X (\omega ),B )$ is a version of $ P_{\sigma(X)} \{Y \in B \} $, the conditional distribution of $Y $ given $\sigma (X )$
Since the conditional distribution of $Y $ give $\sigma (X )$ is defined as any transition kernel $L $ from $(\Omega , \sigma (X ))$ into $(E, \mathcal E )$ such that for any $\omega \in \Omega $ and any $B \in \mathcal E $, $L (\omega ,B )$ is a version of $ P_{\sigma(X)} \{Y \in B \} $ and and it is clear that $K(X,\cdot )$ is a transition kernel $L $ from $(\Omega , \sigma (X ))$ into $(E, \mathcal E )$ what remains to show is that for any $H \in \sigma (X ) $, $B \in \mathcal E $
$$\int \mathbb P(d \omega ) I _H(\omega )K (X (\omega ),B )=\int \mathbb P (d \omega ) I _H(\omega )Y (\omega )$$
but here I get stuck. Any advice is appreciated!