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!
Suppose $\mu$ is finite. Let $A_n = \{ x | f(x) \in [{1 \over n},0] \}$. Since $f(x) >0$ ae. [$\mu$], we see that $\mu A_n \to 0$. Choose $n$ such that $\mu A_n < { 1\over 2} \epsilon$.
Suppose $\mu E \ge \epsilon$, then $\mu (E \setminus A_n) \ge {1 \over 2 } \epsilon$ and $\int_E f d \mu \ge {1 \over n} {1 \over 2 } \epsilon$.