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

Result R3515 on D1747: Unsigned basic integral

Unsigned basic integral zero iff function almost everywhere zero

Formulation 0

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

(i)	$f : X \to [0, \infty]$ is an D5610: Unsigned basic Borel function on $M$

Then \begin{equation} f \overset{a.e.}{=} 0 \quad \iff \quad \int_X f \, d \mu = 0 \end{equation}

Formulation 1

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

(i)	$f : X \to [0, \infty]$ is an D5610: Unsigned basic Borel function on $M$

Then \begin{equation} f \overset{a.e.}{=} 0 \quad \iff \quad \mu(f) = 0 \end{equation}

Subresults

▶	R3517: Unsigned basic expectation zero iff random number almost surely zero