- #1
Chris L T521
Gold Member
MHB
- 913
- 0
Here's this week's problem!
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Problem: Let $h$ and $g$ be integrable functions on $X$ and $Y$, and define $f(x,y)=h(x)g(y)$. Show that if $f$ is integrable on $X\times Y$ with respect to the product measure, then
\[\int_{X\times Y} f\,d(\mu\times\nu) = \int_X h\,d\mu \int_Y g \,d\nu.\]
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Note: We do not need to assume that $\mu$ and $\nu$ are $\sigma$-finite.
Remember to read the http://www.mathhelpboards.com/showthread.php?772-Problem-of-the-Week-%28POTW%29-Procedure-and-Guidelines to find out how to http://www.mathhelpboards.com/forms.php?do=form&fid=2!
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Problem: Let $h$ and $g$ be integrable functions on $X$ and $Y$, and define $f(x,y)=h(x)g(y)$. Show that if $f$ is integrable on $X\times Y$ with respect to the product measure, then
\[\int_{X\times Y} f\,d(\mu\times\nu) = \int_X h\,d\mu \int_Y g \,d\nu.\]
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Note: We do not need to assume that $\mu$ and $\nu$ are $\sigma$-finite.
Remember to read the http://www.mathhelpboards.com/showthread.php?772-Problem-of-the-Week-%28POTW%29-Procedure-and-Guidelines to find out how to http://www.mathhelpboards.com/forms.php?do=form&fid=2!