ThmDex – An index of mathematical definitions, results, and conjectures.
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}
Subresults
R3512: Unsigned basic integral over set of measure zero is zero
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. $\square$