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

Unsigned basic expectation is compatible with probability measure

Formulation 1

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

(i)	$E \in \mathcal{F}$ is an D1716: Event in $P$
(ii)	$I_E : \Omega \to \{ 0, 1 \}$ is the D41: Indicator function on $\Omega$ with respect to $E$

Then \begin{equation} \int_{\Omega} I_E \, d \mathbb{P} = \mathbb{P}(E) \end{equation}

Formulation 1

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

(i)	$E \in \mathcal{F}$ is an D1716: Event in $P$
(ii)	$I_E : \Omega \to \{ 0, 1 \}$ is the D41: Indicator function on $\Omega$ with respect to $E$

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

Proofs

Proof 0

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

(i)	$E \in \mathcal{F}$ is an D1716: Event in $P$
(ii)	$I_E : \Omega \to \{ 0, 1 \}$ is the D41: Indicator function on $\Omega$ with respect to $E$

This result is a particular case of R1242: Unsigned basic integral is compatible with measure. $\square$