ThmDex – An index of mathematical definitions, results, and conjectures.
Result R3896 on D747: Idempotent map
Idempotence of real expectation
Formulation 0
Let $X \in \text{Random}(\mathbb{R})$ be a D3161: Random real number such that
(i) \begin{equation} \mathbb{E} |X| < \infty \end{equation}
Then \begin{equation} \mathbb{E}(\mathbb{E} X) = \mathbb{E} X \end{equation}
Proofs
Proof 0
Let $X \in \text{Random}(\mathbb{R})$ be a D3161: Random real number such that
(i) \begin{equation} \mathbb{E} |X| < \infty \end{equation}
This result is a particular case of R5302: Idempotence of complex expectation. $\square$