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I wonder if someone can help me solve this question (it's not homework):

Let $(X,\mathcal{A},\mu)$ be a measure space and let $f,h\in \mathcal{L}^1(X,\mathcal{A},\mu,\mathbb{C})$ be integrable functions.

Let $g: X \times X \to \mathbb{C}$ be defined by $g(x,y)=f(x)h(y)$.

Prove that $g$ is $\mathcal{A}\times\mathcal{A}$ measurable.

lioness99a
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user202542
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  • Please check my edit. You "defined" $g:X\times X\to\mathbb C$ by means of some $g:X\to\mathbb C$. That cannot be correct of course. – drhab Mar 13 '17 at 15:36
  • Yes, thanks! Actually I meant to write $g(x,y)=f(x)f(y)$ but they way it's now is more general. – user202542 Mar 13 '17 at 16:07

1 Answers1

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The product $\mu:\mathbb{C}\times \mathbb{C}\rightarrow \mathbb{C}$ defined by $\mu(x,y)=xy$ is measurable. The function $h:X\times X\rightarrow \mathbb{C}\times \mathbb{C}$ defined by $h(x)=(f(x),g(x))$ is measurable. This implies that $\mu\circ h$ is measurable.

Tsemo Aristide
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