ThmDex – An index of mathematical definitions, results, and conjectures.
Conditional expectation of known random real number
Formulation 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) $X : \Omega \to \mathbb{R}$ is a D3161: Random real number on $P$
(iii) \begin{equation} \mathbb{E} |X| < \infty \end{equation}
(iv) \begin{equation} X \in \mathcal{G} \end{equation}
Then \begin{equation} \mathbb{E}(X \mid \mathcal{G}) \overset{a.s.}{=} X \end{equation}
Formulation 1
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) $X : \Omega \to \mathbb{R}$ is a D3161: Random real number on $P$
(iii) \begin{equation} \mathbb{E} |X| < \infty \end{equation}
(iv) \begin{equation} \sigma_{\text{pullback}} \langle X \rangle \subseteq \mathcal{G} \end{equation}
Then \begin{equation} \mathbb{P} \left( \mathbb{E}(X \mid \mathcal{G}) = X \right) = 1 \end{equation}
Proofs
Proof 1
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) $X : \Omega \to \mathbb{R}$ is a D3161: Random real number on $P$
(iii) \begin{equation} \mathbb{E} |X| < \infty \end{equation}
(iv) \begin{equation} X \in \mathcal{G} \end{equation}
This result is a particular case of R2160: Conditional expectation of known random complex number. $\square$