Show that if f is a measurable real-valued function and g is a continuous function defined on , then is measurable.
Please help! :'(
This is what I have developed in a moment of clarity; Please tell me if I am right or wrong:
Certainly every continuous function is measurable, since every interval is measurable (P. 67). Therefore, is measurable.
Since is a real-valued function, restrict from to the co-domain of , which is a subset of . Since is measurable, the domain of restricted to the co-domain of has a measurable domain, that is, the set of all .
Also, the set of all elements not in the domain of are , which must neccessarily be measurable.
Therefore, since is measurable and , the domain of , and both and are measurable, then by Problem 21.a., the function restricted to the co-domain of must be measurable. Then is measurable since for all , an element in the co-domain of is defined, and restricted to these elements is measurable.