Let $X$ be a locally compact, Hausdorff, topological space and denote by $\mathcal B_+(X)$ the collection of all Borel-measurable functions from $X$ to $[0,+\infty]$ (extended positive reals).

Suppose that we are given a map $$ I:\mathcal B_+(X) \to [0,+\infty] $$ which is positively homogeneous and satisfies $$ I\big(\sum_{n=1}^\infty f_n\big) = \sum_{n=1}^\infty I(f_n), $$ for every sequence $\{f_n\}_n$ in $\mathcal B_+(X)$.

Observe that this is essentially the Daniell integral.

Suppose further that $\mu$ is a given regular Borel measure on $X$ such that $$ I(f) = \int_X f\,d\mu, $$ for every non-negative $f$ in $C_c(X)$, the space of all compactly supported continuous functions on $X$. Does it follow that this identity also holds for every $f$ in $\mathcal B_+(X)$?

Ok to assume that $X$ is metrizable and $\sigma$-compact, if necessary.