Q:

Using Riesz Representation Theorem to find positive linear functionals

I am having trouble using the Riesz representation theorem in concrete examples. For instance, I came upon this problem (The RRT is the one in Chapter 2 of Papa Rudin and I preserve his notation):
Find a positive linear functional $\phi$ on $C_c([\frac{\pi}{16}, 2 \pi])$ such that $$ \phi(f)=\phi(g)=1,$$ where $f(x)= \frac{\sin x}{x}$ and $g(x)=\frac{\cos x}{x}.$
I think I would know how to attack problems that give me the linear functional but not the measure, but since on the proof of RRT we start with a given functional $\Lambda$, I am at a loss on how to begin thinking about this.

A:

There's a wide choice of possible answers. For any finite Borel measures $\mu$ and $\nu$ on $[\pi/16, 2\pi]$ such that the vectors $V_1 = [\int f \; d\mu, \int g \; d\mu]$ and $V_2 = [\int f \; d\nu, \int g \; d\nu]$ are linearly independent, take $\phi$ corresponding to 
the measure $\alpha \mu + \beta \nu$ where $\alpha V_1 + \beta V_2 = [1,1]$.

