Suppose $\{f_n\}$ are real valued nonnegative continuous functions on $\bar\Omega$, where $\Omega$ is a bounded open subset of $\mathbb{R}^N$. Moreover, say $\int_\Omega f_n\le 1$ for all n. Then $\{f_n\}$ are bounded in $[L^\infty(\Omega)]^*$ (the dual of $L^\infty(\Omega)$) and $[C(X)]^*$ (the dual of $C(\bar\Omega)$). Right?
If we apply Riesz representation theorem for $[L^\infty(\Omega)]^*$ and Alaoglu theorem, $f_n$ has a subsequence that converges to a bounded finite additive measure $\mu$ that is absolutely continuous with respect to the Lebesgue measure. If we apply the representation theorem for $[C(X)]^*$, the subsequence converges to a Randon measure $\nu$. Now, what is the relation of $\mu$ and $\nu$?? Since $\nu$ is countable additive on Borel sets, does that mean $\mu$ is countable additive on Borel sets?
Let's say $\Omega=[0, 1]$ and $f_n$ is affine with $f_n(0)=f_n(\frac{1}{n})=0$ and $f_n(\frac{1}{2n})=2n$ so that $\int_0^1f_n=1$ for each $n$. Question: what is the limit of $f_n$ in $[C(X)]^*$ and $[L^\infty(\Omega)]^*$?