ThmDex – An index of mathematical definitions, results, and conjectures.
Expectation of conditional expectation for a random euclidean real number

Let $P = (\Omega, \mathcal{F}, \mathbb{P})$ be a D1159: Probability space such that
 (i) $\mathcal{G} \subseteq \mathcal{F}$ is a D470: Subsigma-algebra of $\mathcal{F}$ on $\Omega$ (ii) $X : \Omega \to \mathbb{R}^D$ is a D4383: Random euclidean real number on $P$ (iii) $$\mathbb{E} |X| < \infty$$
Then $$\mathbb{E}(\mathbb{E}(X \mid \mathcal{G})) = \mathbb{E}(X)$$
Subresults
 ▶ R5170: Expectation of conditional expectation for a random complex number ▶ R3649: Expectation of conditional probability
Proofs
Proof 0
Let $P = (\Omega, \mathcal{F}, \mathbb{P})$ be a D1159: Probability space such that
 (i) $\mathcal{G} \subseteq \mathcal{F}$ is a D470: Subsigma-algebra of $\mathcal{F}$ on $\Omega$ (ii) $X : \Omega \to \mathbb{R}^D$ is a D4383: Random euclidean real number on $P$ (iii) $$\mathbb{E} |X| < \infty$$
Since $\mathcal{G}$ is a sigma-algebra on $\Omega$, then $\Omega \in \mathcal{G}$. Since $\mathbb{E}(X \mid \mathcal{G})$ is a conditional expectation of $X$ given $\mathcal{G}$ and since $\Omega \in \mathcal{G}$, then $$\mathbb{E}(\mathbb{E}(X \mid \mathcal{G})) = \mathbb{E}(\mathbb{E}(X \mid \mathcal{G}) I_{\Omega}) = \mathbb{E}(X I_{\Omega}) = \mathbb{E}(X)$$ $\square$