Conditional expectation from finite-measure densities (${\int }_A{f}_{0}d\mu ={\int }_{A}fd\mu (A\in {\Sigma }_0)$)

For nonnegative integrable $f$, the finite measure $A\mapsto {\int }_{A}fd\mu$ restricted to a sub-sigma-algebra ${\Sigma }_0$ is absolutely continuous with respect to the restricted base measure. Its Radon-Nikodym derivative is ${\Sigma }_0$-measurable and has the required integral identities. Positive and negative parts extend the construction to real integrable functions; real and imaginary parts extend it to complex functions. The result is unique almost everywhere.