Does $\int_a^bf^p=0$ imply $\int_a^bf=0$?

Suppose $ 0<p<1$. Let $M=\sup \{f(x): a \leq x \leq b\}$. Let $g=\frac f M$. Then $\int g^{p}=0$ and $0\leq g \leq 1$. Hence $0 \leq g \leq g^{p}$ which gives $0\leq \int g \leq \int g^{p}=0$ which gives $\int g=0$. This implies $\int f=0$.