If the Lebesgue integral of a strictly positive function is zero...

Question

If the Lebesgue integral (over a set A) of a strictly positive function is zero, it means that the Lebesgue measure of A is zero?

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Answer 1

Since $f \chi_A \ge 0$ you can use Chebyshev's inequality. For any $\epsilon > 0$ you have $$\mu(\{f \chi_A > \epsilon\}) \le \frac{1}{\epsilon} \int f \chi_A \, d\mu = 0.$$ Take the union over a sequence $\epsilon_k \searrow 0$ to obtain $\mu(\{f \chi_A > 0\}) = 0$. Finally note $$x \in A \implies f(x) \chi_A(x) > 0 \implies x \in \{f \chi_A > 0\}$$ so that $A \subset \{f \chi_A > 0\}$. Thus $\mu(A) = 0$.