## Transition kernel

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

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

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

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

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

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