Consider the $\displaystyle \sigma$-algebra $\displaystyle B:=B(I)$ of Borel of $\displaystyle I:= [0,1]$. $\displaystyle B(\mathbb{R})$ the Borel of $\displaystyle \mathbb{R}$ We say that a function $\displaystyle Q:I \times B \longrightarrow{I}$ is a transition kernel if:

For each $\displaystyle x\in I$ $\displaystyle Q(x,.)$ is a probability measure on B.
For each $\displaystyle A\in B$ $\displaystyle Q(.,B)$ is a function measurable B-B.

a) Prove that if $\displaystyle f:I \times I\longrightarrow{\mathbb{R_+}}$ is $\displaystyle B\otimes B-B(\mathbb{R}) $ measurable, then the function $\displaystyle x\rightarrow{\int\limits_I {f(x,y)Q(x,dy)}}$ defined for each $\displaystyle x\in{I}$ is $\displaystyle B-B(\mathbb{R})$ measurable.

Suppose further that $\displaystyle \int\limits_I {f(x,y)dx = 1}$ for each $\displaystyle y\in{I}$:

b) Prove that $\displaystyle (y,A)\rightarrow{\int\limits_I {f(x,y)Q(x,B)dx = 1}}$ defined on $\displaystyle I \times B$ is a transition kernel.