Let $f$ and $g$ two measurable functions defined on $X$ and $Y$ measure space, with complex values. I have to prove that the function $f(x-y)g(y)$, defined on $X\times Y$, is measurable with respect to the product sigma-algebra.
Measurable function on $X\times Y$
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analysis
measure-theory
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1 Answers
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Hint: $X$ and $Y$ have to be subspaces of a vector space for $x-y$ to make sense.
The function defined $h$ on $X\times Y$ by $h(x,y)=x-y$ is continuous thus mesurable, so $f'(x,y)=f(x-y)$ is measurable since it is the composition of meaasurable functions.
$(f',g):X\times Y\rightarrow C\times C$ is measurable since the product of measurable functions is measurable
The product $C\times C\rightarrow C$ is measurable. You can write $f(x-y)g(y)$ as a composition of measurable functions, so it is measurable.