How to show that if $\int_a^b|{f(x)}|^p\,d\alpha = 0$ then $\int_a^b|{f(x)}|\,d\alpha = 0$ How can I prove that if $\int_a^b|{f(x)}|^p\,d\alpha = 0$ then $\int_a^b|{f(x)}|\,d\alpha = 0$? (Assuming that $\alpha$ is increasing and $p > 1$). Intuitively this is clear but I'm having trouble proving it rigorously.

Let $q$ such that $\dfrac{1}{p}+\dfrac{1}{q}=1$ so $q>1$ and
$$\int_a^b|f|d\alpha\leq\Big(\int_a^b|f|^pd\alpha\Big)^\frac1p\Big(\int_a^b|f|^qd\alpha\Big)^\frac1q=0$$ 
Without Holder, we say for all step function $\phi$
$$\int_a^b|f|^pd\alpha\leq\sup_{|f|^p\leq\phi}\int_a^b\phi d\alpha=0$$
this shows $|f|=0$ almost-everywhere for $\alpha$ (measure) on $[a,b]$. That is
$$\int_a^b|f|d\alpha=0$$
