I am trying to solve the following Lebesgue integral problem:
Suppose $\mu(X)$ is finite. Let $f$ be an integrable function such that $f>0$ almost everywhere. For any $\varepsilon > 0$, there is a $\lambda>0$ such that $$\int_E f \,d\mu \geq \lambda$$ for all measurable $E$ with measure $\mu(E) \geq \varepsilon$
This question seems really intuitive in that set of nonnegative measure will have nonnegative integral, but I really have no idea where to start. Thank!