A measurable function on a measure subspace is a function that maps a set of measurable sets onto a measure space, such that the pre-images of measurable sets are also measurable sets.
A measurable function is defined in terms of measurable sets and their pre-images, while a continuous function is defined in terms of open sets and their pre-images. Measurable functions are more general than continuous functions, as they do not require the use of topology.
A function is measurable on a measure subspace if the pre-image of any measurable set in the codomain is a measurable set in the domain. This can be verified by checking the pre-images of a set of generating measurable sets.
Yes, a non-measurable function can exist on a measure subspace. This can occur if the measure subspace does not contain enough generating measurable sets to fully characterize the function. However, this is a rare occurrence and most functions on measure subspaces are measurable.
Measurable functions on measure subspaces are important in the field of real analysis as they allow for the integration of more general functions. They also play a crucial role in the development of the Lebesgue measure and integral, which are widely used in mathematics and physics.