Category:Integrable Function with Zero Integral on Sub-Sigma-Algebra is A.E. Zero

This category contains pages concerning Integrable Function with Zero Integral on Sub-Sigma-Algebra is A.E. Zero:

Let $\struct {X, \Sigma, \mu}$ be a measure space.

Let $\GG$ be a sub-$\sigma$-algebra of $\Sigma$.

Let $f : X \to \overline \R$ be a $\GG$-integrable function.

Suppose that, for all $G \in \GG$:

$\ds \int_G f \rd \mu = 0$

Then $f = 0$ $\mu$-almost everywhere.