That seems like an odd definition of measurability to me; are you sure that's right? Usually one would require that the inverse image of open (or measurable) sets be measurable sets, and it suffices to check that is measurable for all assuming the Borel algebra on (see Real and Complex Analysis by Rudin, for example).
Anyways, I don't think either is measurable because you don't get measurable sets after taking the inverse image of and respectively. A measurable map is, for example, one that takes everyone to .
To get the kind of F you want, double click on this: .