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

Result R4562 on D1748: Signed basic integral

Basic integral over a set of measure zero is zero

Formulation 0

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

(i)	$f : X \to [- \infty, \infty]$ is a D5600: Basic Borel function on $M$
(ii)	$E \in \mathcal{F}$ is a D1109: Measurable set in $M$
(iii)	\begin{equation} \mu(E) = 0 \end{equation}

Then \begin{equation} \int_E f \, d \mu = 0 \end{equation}

Proofs

Proof 0

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

(i)	$f : X \to [- \infty, \infty]$ is a D5600: Basic Borel function on $M$
(ii)	$E \in \mathcal{F}$ is a D1109: Measurable set in $M$
(iii)	\begin{equation} \mu(E) = 0 \end{equation}

Since we define \begin{equation} \int_E f \, d \mu = \int_E f^+ \, d \mu - \int_E f^- \, d \mu \end{equation} then this result follows as a consequence of the result R3512: Unsigned basic integral over set of measure zero is zero. $\square$