Thread: 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 ??

Any ideas please ?? Thanks

Can you give information about the set on which and are defined?

It is R the real numbers. I think I mentioned that in the question. Thank you !

Do you mean Borel or Lebesgue measurable?

Lebesgue measurable