Let $\displaystyle (S,\Sigma,\mu)$ be a measure space. Prove that if $\displaystyle f$ is a Lebesgue integrable function, then for each $\displaystyle \varepsilon > 0$ there exists a $\displaystyle \delta > 0$ such that $\displaystyle \int_A \mid f \mid \mathrm{d} \mu$ for each $\displaystyle A \in \Sigma$ with $\displaystyle \mu(A) < \delta$.