A remark on conditional expectations (Q917134)

Let (\(\Omega\),\({\mathcal F},P)\) be a probability space, \({\mathcal G}\subseteq {\mathcal F}\) a sub \(\sigma\)-algebra and X an \({\mathcal F}\)-measurable random variable with values in [-\(\infty,\infty]\) which is not supposed to be integrable. The author defines E(X \(| {\mathcal G}):=E(X^+ | {\mathcal G})-E(X^- | {\mathcal G})\) if the right-hand side is not of the form \(\infty - \infty\) and derives the well-known rules for handling the conditional expectation.