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

Unsigned basic expectation over set of probability zero is zero

Formulation 0

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

(i)	$\mathbb{P}(E) = 0$ for $E \in \mathcal{F}$

Let $X : \Omega \to [0, \infty]$ be a D5101: Random unsigned basic number on $P$.

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

Formulation 1

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

(i)	$\mathbb{P}(E) = 0$ for $E \in \mathcal{F}$

Let $X : \Omega \to [0, \infty]$ be a D5101: Random unsigned basic number on $P$.

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

Proofs

Proof 0

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

(i)	$\mathbb{P}(E) = 0$ for $E \in \mathcal{F}$

Let $X : \Omega \to [0, \infty]$ be a D5101: Random unsigned basic number on $P$.

This result is a particular case of R3512: Unsigned basic integral over set of measure zero is zero. $\square$