- #1
Chris L T521
Gold Member
MHB
- 913
- 0
Here's this week's problem!
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Problem: Let $g:\mathbb{R}\rightarrow\mathbb{R}$ be a mapping of $\mathbb{R}$ onto $\mathbb{R}$ for which there is a constant $c>0$ for which
\[|g(u)-g(v)|\geq c|u-v|\text{ for all $u,v\in\mathbb{R}$.}\]
Show that if $f:\mathbb{R}\rightarrow\mathbb{R}$ is Lebesgue measurable, then so is the composition $f\circ g:\mathbb{R}\rightarrow\mathbb{R}$.
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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 $g:\mathbb{R}\rightarrow\mathbb{R}$ be a mapping of $\mathbb{R}$ onto $\mathbb{R}$ for which there is a constant $c>0$ for which
\[|g(u)-g(v)|\geq c|u-v|\text{ for all $u,v\in\mathbb{R}$.}\]
Show that if $f:\mathbb{R}\rightarrow\mathbb{R}$ is Lebesgue measurable, then so is the composition $f\circ g:\mathbb{R}\rightarrow\mathbb{R}$.
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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!