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

Processing math: 100%

Result R5300 on D1747: Unsigned basic integral

Unsigned basic integral over an empty set equals zero

Formulation 0

Let

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

(1)	$f : X \to [0, \infty]$ is a D313: Measurable function on $M$

Then

$\begin{equation} \int_{\emptyset} f \, d \mu = 0 \end{equation}$

Formulation 1

Let

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

(1)	$f : X \to [0, \infty]$ is a D313: Measurable function on $M$

Then

$\begin{equation} \int_X I_{\emptyset} f \, d \mu = 0 \end{equation}$

Subresults

▶	R5301: Random unsigned basic number has zero correlation with the empty indicator

Proofs

Proof 0

Let

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

(1)	$f : X \to [0, \infty]$ is a D313: Measurable function on $M$

Since

$x \not\in \emptyset$ for all

$x \in X$ , then we have

$I_{\emptyset}(x) f(x) = 0$ for all

$\begin{equation} \int_{\emptyset} f \, d \mu = \int_X I_{\emptyset} f \, d \mu = 0 \end{equation}$

$\square$