= Conditional expectation from finite-measure densities
{title2=$\int_A f_0\,d\mu=\int_Af\,d\mu\quad(A\in\Sigma_0)$}
For nonnegative <integrable> $f$, the finite <measure> $A\mapsto\int_Af\,d\mu$ 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>.
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