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

Unsigned basic integral is compatible with measure

Formulation 0

Let $M = (X, \mathcal{F}, \mu)$ be a D1158: Measure space such that

(i)	$E \in \mathcal{F}$ is a D1109: Measurable set in $M$
(ii)	$I_E : X \to \{ 0, 1 \}$ is an D41: Indicator function on $M$ with respect to $E$

Then \begin{equation} \mu(E) = \int_X I_E \, d \mu \end{equation}

Formulation 1

Let $M = (X, \mathcal{F}, \mu)$ be a D1158: Measure space such that

(i)	$E \in \mathcal{F}$ is a D1109: Measurable set in $M$

Then \begin{equation} \mu(E) = \int_E \, d \mu \end{equation}