# measurable functions

• Nov 28th 2011, 07:36 PM
hgd7833
measurable functions
Let f be a measurable function and g be a 1-1 function from R to R which has a Lipschitz inverse.

Show that the composition fog is measurable

My idea was to take a set (a,00) , then f^-1(a,00) is measurable call this set C , but i need to prove that g^-1(C) is measurable
Since g^-1 is Lipschitz then it is continuous. Let g^-1 = h , so how can I prove that h(C) is measurable provided that h is Lipschitz, hence continuous ??

• Nov 30th 2011, 03:48 AM
girdav
Re: measurable functions
Can you give information about the set on which $f$ and $g$ are defined?
• Dec 2nd 2011, 08:13 AM
hgd7833
Re: measurable functions
It is R the real numbers. I think I mentioned that in the question. Thank you !
• Dec 2nd 2011, 09:57 AM
girdav
Re: measurable functions
Do you mean Borel or Lebesgue measurable?
• Dec 3rd 2011, 05:29 PM
hgd7833
Re: measurable functions
Lebesgue measurable