ThmDex – An index of mathematical definitions, results, and conjectures.

Result R5502 on D2280: Conditional expectation

Real conditional expectation with respect to a constant random real number

Formulation 0

Let $X, Y \in \text{Random}(\mathbb{R})$ be a D3161: Random real number such that

(i)	\begin{equation} \exists \, y \in \mathbb{R} : Y = y \end{equation}

Then \begin{equation} \mathbb{E} (X \mid Y) \overset{a.s.}{=} \mathbb{E} X \end{equation}

Proofs

Proof 0

Let $X, Y \in \text{Random}(\mathbb{R})$ be a D3161: Random real number such that

(i)	\begin{equation} \exists \, y \in \mathbb{R} : Y = y \end{equation}

We define $\mathbb{E}(X \mid Y)$ as $\mathbb{E}(X \mid \sigma_{\text{pullback}} \langle Y \rangle)$. Hence, this result is a consequence of the results

(i)	R5504: Constant random variable pulls back a bottom sigma-algebra
(ii)	R2522: Real conditional expectation given minimal information

$\square$