Hi, it's been awhile since I have studied Lebesgue measure so I'm trying to re-learn the material on my own. Most of my friends don't remember much as well so it's been a bit of a struggle trying to work on these problems on my own. Thank you for any kind of help!(adsbygoogle = window.adsbygoogle || []).push({});

OMIT Question 1.If [itex]f:[a,b]\rightarrow \mathbb{R} [/itex] is absolutely continuous and one-to-one, then is [itex]f^{-1}[/itex] absolutely continuous? If so, prove. If not, provide a counterexample.

OMIT Question 2.If [itex]f:[a,b]\rightarrow [c,d] [/itex] is 1-1 and absolutely continuous, then

OMIT a.is [itex]f^{-1}[/itex] of bounded variation on [itex][c,d] [/itex]?

OMIT b.if [itex]E \subseteq [c,d] [/itex] and [itex]m(E)=0[/itex], then do you think [itex]m(f^{-1}(E))=0[/itex]? If I know that [itex]f^{-1}[/itex] is absolutely continuous, then I can prove that [itex]f^{-1}[/itex] sends sets of measure zero to sets of measure zero....

Question 3.Let [itex] f:[0,1] \rightarrow\mathbb{R} [/itex] satisfy [itex] |f(x) - f(y)|\leq |x^{1/3}-y^{1/3}| [/itex] for all [itex]x,y \in [0,1][/itex]. Must [itex]f [/itex] be absolutely continuous? Justify.

My question 3 is hard....

Edit:I removed the last problem which I eventually figured out...

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