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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.

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    Why is it relevant that the measure space is a product measure space?2012-12-10
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    I tried to prove the inverse image of every open set is measureable, but I found it is difficult unless I assume $\sigma$-finiteness, etc.2012-12-10
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    You need to be more specific with what you're asking. What are the domains and codomains for your functions? You should edit your question to include this. For example, if $f,g \colon X \to \mathbb{R}$, where $X$ is an arbitrary measure space and $\mathbb{R}$ has the Borel $\sigma$-algebra, then $fg$ is measurable. The fact that you're looking at a product space would be irrelevant here. Presumably there's something about your question in particular which makes it relevant.2012-12-10
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    We assume $X,Y$ to be two arbitrary given measurable space. And $(X\times Y)$ were given the product measure.2012-12-10
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    When you say measurable function then you mean a real-valued function? Such that the preimage of an open set is measurable? Or are you interested in the general topological setting?2012-12-10
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    Yes. I mean a measurable function with real or complex value.2012-12-10
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    Got it. thanks.2012-12-10

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