I'm reading about this in one of the articles that the teacher gave me
Suppose that and are two -finite measure spaces and is measurable. Then
with obvious modifications in the case . If , and both sides are finite, then equality holds only if a.e. for some non-negative measurable functions and .
If is non-negative, then the function
and thus are measurable. For the LHS to be well-defined, the function should be measurable. In our case, is not assumed to be non-negative. Could you elaborate on how to prove it?