How can I prove the product of two measurable functions in the product measure space is measurable? I tried but still do not know how to prove.
Question in product measure
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measure-theory
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0Got it. thanks. – 2012-12-10
1 Answers
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My proof:
It suffice to prove that $f_{1}(x,y)=g(x)$ and $f_{2}(x,y)=h(y)$ are measurable functions in $(\mathcal{S}\times \mathcal{T})$. Then we should have $f(x,y)=f_{1}(x,y)*f_{2}(x,y)$. Since the product of measurable functions is measurable, we concluded the proof.
By assumption $g(x),h(y)$ are both measurable functions in coordinates. Now for any open set $O$, $f_{1}^{-1}(O)$ is equal to $g^{-1}(O)\times \mathcal{T}$, similarly $f_{2}^{-1}(O)$ is equal to $h^{-1}(O)\times \mathcal{S}$. Both sets are measurable sets by definition of product measure (Tao, page 195). Therefore both $f_{1},f_{2}$ are measurable.
We conclude that $f$ must be measurable.