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Let $f:[a,b] \rightarrow \mathbb{R}$ be a Lipschitz function. I want to show that it carries $F_\sigma$ sets to $F_\sigma$ sets.

I'm not sure how to demonstrate this. Specifically I'm not sure what property of continuity or Lipschitz would preserve the $F_\sigma$ property. I do know that this is true: $f(\bigcup_{i} A_i)=\bigcup_{i}f(A_i)$.

emka
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1 Answers1

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Hint: A closed set $K \subseteq [a,b]$ is compact (as $[a,b]$ is). Hence its image $f[K]$ under the continuous function $f$ is compact also.

martini
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  • Does it just immediately follow that $f(\bigcup_{i} K_i)=\bigcup_{i}f(K_i)$ since subsets of compact sets are compact? – emka Oct 04 '12 at 11:00
  • You said you know about this. It holds $x \in f[\bigcup_i K_i] \iff \exists y \in \bigcup_i K_i. f(y) = x $ $\iff \exists i\exists y \in K_i. f(y) = x \iff \exists i. x \in f[K_i] \iff x\in \bigcup_i f[K_i]$. – martini Oct 04 '12 at 11:08
  • I think it was just stated as a definition in Principles of Analysis by Rudin. I didn't know (or most likely cannot recall) that it works for compactness. – emka Oct 04 '12 at 11:14
  • The statement $f[\bigcup_i K_i] = \bigcup_i f[K_i]$ has nothing to do with the compactness of the $K_i$. It just follows from the definition of union and image. – martini Oct 04 '12 at 11:24
  • I'm sorry...I don't follow. Compact sets are closed and bounded and preserved by a continuous function f. However, $F_\sigma$ just have the property of being a countable union of closed sets. – emka Oct 04 '12 at 11:32
  • Let $A$ be an $F_\sigma$-set. Then $A$ can be written as $A = \bigcup_{i\ge 1} K_i$ with compact(!) $K_i$. Now the $f[K_i]$ are compact, hence closed, so $f[A] = f[\bigcup_i K_i] = \bigcup_i f[K_i]$ is a ..... – martini Oct 04 '12 at 11:37
  • Why is a closed set the countable union of compact sets? – emka Oct 04 '12 at 11:49
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    $A$ is an $F_\sigma$-set in $[a,b]$. Hence it's the union of countable many closed $K_i \subseteq [a,b]$. As $[a,b]$ is compact, each of it's closed subsets is also. So $A$ is a countable union of compact sets. – martini Oct 04 '12 at 11:51