Let $\alpha$ be a monotonically increasing continuous function. Suppose $f \in C[a,b]$ satisfies the property $$\int fg \; d\alpha=0, \forall g \in C[a,b]$$ Then show that $f=0$
This is how I tried: Assume not. Then f is non zero at least one point. Consider a interval around that point. f is non zero at that interval, wlog, say positive. Then I need to arrive at a contradiction!
I can also show there is one point in $[a,b]$ where $f(c)=0$ using intermediate value theorem for Stieltjes integral. I don't know how to show whole of f is zero from here
You are on the right track. Assume as you do that $f(x_0)>0$ at some point $x_0$. Then it's positive in an interval - by continuity. Next choose $g$ to be some continuous function on that interval, such that g(x)=0 outside the interval and positive inside the interval. Since $\alpha $ is increasing we have that your integral is positive. Hence we have arrived at a contradiction. Thus $f(x_0)$ had to be 0.