Let $\displaystyle f(x)$ be continuous on $\displaystyle [a,b]$. Let $\displaystyle \alpha (x)$ be increasing on $\displaystyle [a,b]$.
Suppose $\displaystyle \int f(x)g(x)d\alpha (x) \equiv 0$ for every $\displaystyle g(x)$ such that $\displaystyle f(x)g(x) \in R(\alpha)$.
Prove $\displaystyle f(x) \equiv 0$ on $\displaystyle [a,b]$
2. Take $\displaystyle g(x)=f(x)$ on $\displaystyle [a,b]$. Then you just need to show that if $\displaystyle w(x)\ge{0}$, continuous, and $\displaystyle \int_a^bw(x)dx=0$ then $\displaystyle w(x)\equiv{0}$ on the interval.
Hint: Try contradiction, and use the fact that $\displaystyle w$ is continuous.