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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