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

Conditional expectation of a random real number conditioned on itself

Formulation 0

Let $P = (\Omega, \mathcal{F}, \mathbb{P})$ be a D1159: Probability space such that

(i)	$X : \Omega \to \mathbb{R}$ is a D3161: Random real number on $P$
(ii)	\begin{equation} \mathbb{E} \|X\| < \infty \end{equation}

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

Formulation 1

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}(X \mid X) \overset{a.s.}{=} X \end{equation}

Proofs

Proof 0

Let $P = (\Omega, \mathcal{F}, \mathbb{P})$ be a D1159: Probability space such that

(i)	$X : \Omega \to \mathbb{R}$ is a D3161: Random real number on $P$
(ii)	\begin{equation} \mathbb{E} \|X\| < \infty \end{equation}

Since we define $\mathbb{E}(X \mid X) : = \mathbb{E}(X \mid \sigma_{\text{pullback}} \langle X \rangle)$, this result is a special case of R4781: Conditional expectation of known random real number. $\square$