I'm reading tao's book an epsilon of room. I have difficulty understanding a step in the proof of Riesz representation of theorem for $C_c(X\to\mathbf{R})$ unsigned version, the proof can be found here
The only remaining thing to check is that $I(f)=I_\mu(f)$ for all $f\in C(X\to\mathbb{R})$. If $f$ is a finite non-negative linear combination of indicator functions of open sets, the claim is clear from the construction of $\mu$ and additivity of $I$ on $BC_{lst}(X\to\mathbb{R}^+)$; taking uniform limits, we obtain the claim for non-negative continuous functions, and then by linearity we obtain it for all functions.
Here $X$ is a compact Hausdorff space and $\mu$ is a Randon measure on $X$. $I:C_c(X\to\mathbb{R})\to\mathbb{R}$ is a positive linear functional (i.e. $I(f)\geq 0$ whenever $f\geq 0$) and has been extended to all bounded lower semi-continuous non-negative functions $BC_{lst}(X\to\mathbb{R}^+)$.
What does taking uniform limits mean? I guess it means that every non-negative continuous function can be approximated uniformly by finite non-negative linear combination of indicator function of open sets. That is for every non-negative continuous function $f\in C(X\to\mathbb{R}^+)$ and $\varepsilon>0$, there exist a $g=\sum_{i=1}^n c_i1_{U_i}$ ($c_i\geq 0$) such that $$\sup_{x\in X}|f(x)-g(x)|\leq\varepsilon.$$ Here $U_i$ are open sets, I tried to prove this but I cannot work it out.