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

Result R3516 on D5102: Basic expectation

Signed basic expectation of almost surely zero random number is zero

Formulation 0

Let $P = (\Omega, \mathcal{F}, \mathbb{}P)$ be a D1159: Probability space.
Let $X : \Omega \to [-\infty, \infty]$ be an D3066: Absolutely integrable random number on $P$ such that

(i)	$X =_{\mathsf{a.s.}} 0$

Then \begin{equation} \mathbb{E}(X) = 0 \end{equation}

Formulation 1

Let $P = (\Omega, \mathcal{F}, \mathbb{}P)$ be a D1159: Probability space.
Let $X : \Omega \to [-\infty, \infty]$ be an D3066: Absolutely integrable random number on $P$ such that

(i)	$X =_{\mathsf{a.s.}} 0$

Then \begin{equation} \int_{\Omega} X \, d \mathbb{P} = 0 \end{equation}

Proofs

Proof 0

Let $P = (\Omega, \mathcal{F}, \mathbb{}P)$ be a D1159: Probability space.
Let $X : \Omega \to [-\infty, \infty]$ be an D3066: Absolutely integrable random number on $P$ such that

(i)	$X =_{\mathsf{a.s.}} 0$

This result is a particular case of R1903: Basic integral of almost everywhere zero function is zero. $\square$